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","Pid":"PA256","PgNum":"256","Order":269},{"Title":"The Form in which the Elements and Coordinates appear ","Pid":"PA262","PgNum":"262","Order":275},{"Title":"Brief Historical Review of the Theories of Perturbations ","Pid":"PA278","PgNum":"278","Order":291},{"Title":"Notes on the Tables ","Pid":"PA290","PgNum":"290","Order":303}]}},"table_of_contents_page_id":"PP11","max_resolution_image_width":800,"max_resolution_image_height":1242,"num_toc_pages":2,"quality_info":"We think this book has good 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PREFACE . The determination of the motions of the heavenly bodies is an import- ant problem in and for itself , and also on account of the influence it has exerted on the development of \u003cb\u003emathematics\u003c/b\u003e . It has engaged the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PP9\u0026dq=dziobek+mathematical"},{"page_id":"PP11","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. TABLE OF CONTENTS . FIRST DIVISION . SOLUTION OF THE PROBLEM OF TWO BODIES . FORMATION OF THE GEN- ALGEBRAICAL ERAL INTEGRALS FOR THE PROBLEM OF ʼn BODIES . TRANSFORMATIONS OF THIS PROBLEM . PAGE . 1. Newton\u0026#39;s Law of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PP11\u0026dq=dziobek+mathematical"},{"page_id":"PP13","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. TABLE OF CONTENTS . FIRST DIVISION . SOLUTION OF THE PROBLEM OF TWO BODIES , FORMATION OF THE GENERAL INTEGRALS FOR THE PROBLEM OF n BODIES . ALGEBRAICAL TRANSFORMATIONS OF THIS PROBLEM . PAGE . 1 12 19 1. Newton\u0026#39;s Law of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PP13\u0026dq=dziobek+mathematical"},{"page_id":"PP14","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ERRATA . PAGE LINE 10 -6 , -7 FOR motus 11 +8 † 13 headline 17 66 PROBLEMS 66 READ Motus . * PROBLEM 66 21 . +5 91 21 +9 22 +11 fi 22 -4 to 24 +10 27 31 66 3 xz 31 +6 40 +19 M1 42 +15 0 = headline 91 ( x — a + √x ( −2a )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PP14\u0026dq=dziobek+mathematical"},{"page_id":"PA2","page_number":"2","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Since the two particles attract each other , the direction of action is along the line which joins them . The action of P2 on P , has the direction from P1 to P , and its direction cosines are 1 X2 1 2 XC1 Y2 2 r - Yı r Z2\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA2\u0026dq=dziobek+mathematical"},{"page_id":"PA10","page_number":"10","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. because ( 32a ) must give identically r2 = x2 + y2 + z2 . If p is the semi - parameter and e the eccentricity of a conic , its equation , when the origin is at the focus and the + axis is directed to the vertex nearest it\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA10\u0026dq=dziobek+mathematical"},{"page_id":"PA11","page_number":"11","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. the mean distance of the earth from the sun . The sector becomes in a sidereal year the entire area of the ellipse , or S == ab = πανα b2 Vā = πανα Np = = p . If the mean solar day is the unit of time , T is the number of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA11\u0026dq=dziobek+mathematical"},{"page_id":"PA12","page_number":"12","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. 2. ELLIPTIC , PARABOLIC AND HYPERBOLIC ORBITS . Case I , e \u0026lt; 1 . Equation ( 30a ) , § 1 , represents in this case an ellipse . If the semi - axis major is represented by [ a ] , then [ a ] = c2 p 1 2 1 f2 Hence by ( 24 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA12\u0026dq=dziobek+mathematical"},{"page_id":"PA15","page_number":"15","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. 1 II . r = 2 I. p = a cos2 y , Ρ 1 + e cos v \u0026quot; III . r = a ( 1 --- e cos E ) , IV . COS E = cos v + e 1+ e cos v COS E e , or cos v = 1 e cos E V. sin E COS E 1 e = = sin v 2 r ( 1 = sin v e ) = sin v VI . cos E = P. 1+ cos\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA15\u0026dq=dziobek+mathematical"},{"page_id":"PA20","page_number":"20","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. where is again the constant of integration . 2a We will now so select our time that when x = x 。, t = dx \u0026quot; dt to and the velocity , is negative . The planet is then approach- \u0026quot; ing the sun and ( 3 ) becomes dx dt 1 1 √2\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA20\u0026dq=dziobek+mathematical"},{"page_id":"PA24","page_number":"24","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. e3 e ( 4 ) f ( x ) = f ( y ) + — 4 ( y ) ƒ \u0026#39; ( y ) + \u0026quot; 1 3 2 2 ! e * a [ { ( y ) f \u0026#39; ( y ) ] 2 მყ Ə2 [ { + 3 ! ( y ) } 3ƒ \u0026#39; ( y ) ] dy2 + The question of the convergence of this series is one into which we shall not enter\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA24\u0026dq=dziobek+mathematical"},{"page_id":"PA25","page_number":"25","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. e cos E ) may be used . = cos x , and we get To get r , the formula r = a ( 1 In this case , in equation ( 4 ) , f ( x ) r e2 ( 7 ) e cos M + ( 1 α 2 cos 2 M ) e3 2 ! 22 ( 3 cos 3 M ( 42 cos 4M e1 3 ! 23 e5 4 ! 24 ( 53\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA25\u0026dq=dziobek+mathematical"},{"page_id":"PA27","page_number":"27","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. iE iv = iE + loge ( 1 — k [ e ] ̄¡E ) — loge ( 1 — k [ e ] i ) . - Since k \u0026lt; 1 and the modulus of [ e ] iE = 1 , these logarithms can be developed in series , and , passing finally to the trigono- metrical functions , we\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA27\u0026dq=dziobek+mathematical"},{"page_id":"PA28","page_number":"28","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. 1 1 = é cos E , 1 COS E = sin E = 1 = e √1 — e \u0026#39; , hence e 1 ( 12 ) = cos ( M + i No1 e2 ) 2 e 1 arc cos = M + i √1— e2 , or , = cos M cos ( i √1 — e2 ) = cos M [ e ] + [ e ] + VI — es + [ e ] + sin M sin ( i / 1— e3 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA28\u0026dq=dziobek+mathematical"},{"page_id":"PA32","page_number":"32","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. since this integral is always smaller than 2m dE = 2T . Moreover , the roots of J ( x ) = 0 are all real , so that the curve y = J ( x ) is sinuous , like the curve y = sin x , with the difference that the waves for the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA32\u0026dq=dziobek+mathematical"},{"page_id":"PA33","page_number":"33","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. and , for n \u0026gt; 0 , An = 1 2π Ja π ( 1 e cos E ) cos nM dM - 1 π e cos E ) sin n M n 2π 11 ае ηπ 0 2π sin n ( E 247 2π ae 1 n πC sin nM sin E dE e sin E ) sin E dE 1 cos [ ( n - 1 ) E ne sin E ] dE a e n 2π = ae Hence n 1 2π\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA33\u0026dq=dziobek+mathematical"},{"page_id":"PA36","page_number":"36","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Terms of the same form can be obtained for 7 , so that in x only the form given in ( 24 ) can be found . With the aid of ( 22 ) and ( 23 ) , we can at once write out the terms for which [ ^ ] : = α [ ] 1 . They are ( 25 ) a\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA36\u0026dq=dziobek+mathematical"},{"page_id":"PA38","page_number":"38","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. extraordinary degree of learning . This circumstance , together with the religious respect paid in those days to authority , permitted the Almagest to appear as undeniable truth , and any explanation was right or wrong\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA38\u0026dq=dziobek+mathematical"},{"page_id":"PA43","page_number":"43","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. It should be noted that the kinetic energy , and therefore , some velocity , can become infinite , according to ( 14 ) , only when V is infinite , or when one or more distances of the points = 0 . In that case , however\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA43\u0026dq=dziobek+mathematical"},{"page_id":"PA45","page_number":"45","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. To equation ( 17 ) Lagrange gave a form which admits of a highly interesting conclusion concerning the stability of the system . Introducing into ( 4 ) the primed coordinates , and multiplying the equations in order by x\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA45\u0026dq=dziobek+mathematical"},{"page_id":"PA48","page_number":"48","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. plane , and it becomes the invariable plane of the system . This is approximately the case with the solar system . = - If C C C = 0 , every plane is invariable . This would be the case , for instance , if all the points\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA48\u0026dq=dziobek+mathematical"},{"page_id":"PA49","page_number":"49","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. d2x1 dt2 d2x d2 = dt2 dt2 X1 m1 + 3 r1 1 av m1 dx1 X3 m2 3 M3 etc. 3 Introducing finally , Ra = 1 m2 V- x x x 1 + Y λ Y1 +22 21 m1 1 = V Σ m μ ሳን . 3 X 2 X 2 + y 2 Y 2 + z2Z2 3 m2 + z2zμ 3 ru x x x μ + Y z Y μ + zz . Zμ\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA49\u0026dq=dziobek+mathematical"},{"page_id":"PA50","page_number":"50","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 30 ) dy Σαχ Ση dz 2mx 2m ) = C , dt Σ m ( x − y da ) + k ( ≥ m x 2 m dt dt Ση dx Σmy Σmaa ) = C2 , dt dy dt - 1 k Μ + Σm \u0026#39; and 2 2 m dx ́dy ́dz \u0026gt; * [ @ ] + @@ D + CD ] 2 dt dt dy dt 2 dt + 1⁄2 [ ( ± m da ) 2 + ( ≥ m du )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA50\u0026dq=dziobek+mathematical"},{"page_id":"PA54","page_number":"54","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. first order only , instead of the second , and this is more con- venient for many transformations . Equations ( 1 ) retain their form if , instead of the given con- ditions , others are introduced by orthogonal\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA54\u0026dq=dziobek+mathematical"},{"page_id":"PA59","page_number":"59","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 28 ) dt + · [ ş d3 [ 3,3 ] dP3 d3 [ 2,2 ] dt3 dP : — Q. ] m , ru 1 3 r31 1 dt3 dt Q3 ] 3 m3 r12 Besides the integral ( 27 ) , the equations ( 25 ) have another , a combination of the surface integrals , found by Lagrange\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA59\u0026dq=dziobek+mathematical"},{"page_id":"PA61","page_number":"61","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. and only one more relation is needed . Putting η1 tan y = 1 we get ( 34 ) dy = admind_vitri ) ( dỗi + di ) — ( d , tnn ) 2 12 23 € 12 + 712 2 - 2 dr23 123 VICTACTICI 123 12 dt dt The right - hand member is , from the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA61\u0026dq=dziobek+mathematical"},{"page_id":"PA71","page_number":"71","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. in which r1 , r1⁄2 , r are constant and u alone is variable . Intro- ducing these quantities into ( 25 ) , § 7 , we get x2 ( uu ) du du и dt dt A , 1,2 = P + p 3 dt2 И u3 3 1 Here P is what P in ( 26 ) becomes when r1 , T2\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA71\u0026dq=dziobek+mathematical"},{"page_id":"PA75","page_number":"75","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. in the formation of equation ( 34 ) for the determination of p ; this was especially creditable to him because the theory of determinants was then in its infancy . Beyond this point he turned his investigation in the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA75\u0026dq=dziobek+mathematical"},{"page_id":"PA76","page_number":"76","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. noteworthy transformations of the problem of three bodies . Bertrand and Bour ( Mémoire sur le problème des trois corps , Journal de l\u0026#39;école polytechnique , 1856 , p . 35 ) have brought the reduced problem of the eighth\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA76\u0026dq=dziobek+mathematical"},{"page_id":"PA82","page_number":"82","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. i = ni \u0026#39; = n - \u0026gt; i = 1\u0026#39;1 i = ni \u0026#39; = n მ ¢ მს др.да да др . a2ƒ до дф i = ni \u0026#39; = n⋅ деф df მს d2 ¢ of du i = 1 i = 1 i = ni \u0026#39; = n + + Σ i = 1 i = 1 да да др . др i = 1 i = 1 and , as Their alge- The four double\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA82\u0026dq=dziobek+mathematical"},{"page_id":"PA86","page_number":"86","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. as the converse of the other . Lagrange , although he mentions Poisson\u0026#39;s formula , seems to have overlooked this relationship . In order to determine this relationship , we shall collect some proportions in determinants for\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA86\u0026dq=dziobek+mathematical"},{"page_id":"PA94","page_number":"94","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 10 ) дх дх dx = Οπ дл DE \u0026#39; ди θα ди ( 0 % ) ==== It is somewhat more difficult to form the derivatives with respect to e . ( 11 ) we obtain , ( 12 ) From the equations [ == a ( cos E ― e ) 7 = a1e sin E sin E an 1- e cos\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA94\u0026dq=dziobek+mathematical"},{"page_id":"PA96","page_number":"96","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. a1 = a = e agΞΩ a1 = i = a 0 , 0 , [ a , 2 ] , 0 , e 0 , 0 , [ e , 2 ] , 0 , [ e , ] , 0 [ α , π ] , [ α , ε ] 7C Ω ( 19 ) F = - [ a , 2 ] , [ e , 2 ] , 0 , [ 2 , i ] , 0 , 0 i 0 , 0 , [ 2 , i ] , 0 ,\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA96\u0026dq=dziobek+mathematical"},{"page_id":"PA101","page_number":"101","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. partial differential equations between 2n functions of 2n vari- ables , certain fixed conditions must be satisfied . These condi- tions are here satisfied and in such a manner that the solution possesses the greatest\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA101\u0026dq=dziobek+mathematical"},{"page_id":"PA102","page_number":"102","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. and the others can be so selected as to be independent of 7 . For , if we put comes ( 13 ) да Οτ = 0 , the differential equation 0 = ( a , H \u0026#39; ) be- - да 0 = ( a , H ) + ǝt \u0026#39; This equation is consistent , since its\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA102\u0026dq=dziobek+mathematical"},{"page_id":"PA104","page_number":"104","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Therefore , if b , taken as a function of P1 , satisfies no differ- ential equation of the ( 2n - 2 ) th or lower order which contains a and b only in its coefficients , we get from ( 19 ) ( 2n - 2 ) new integrals which\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA104\u0026dq=dziobek+mathematical"},{"page_id":"PA111","page_number":"111","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. of any two functions a ,, a2 of p , q , remains unchanged in form , when p , q , are subjected to a canonical substitution . What precedes is a preparation for our problem , the com- plete integration of the partial\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA111\u0026dq=dziobek+mathematical"},{"page_id":"PA116","page_number":"116","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. and apply to it the so - called contact transformation of p , q , into new variables Pi , Qi , ( 2 ) P ( P1 , ... In ) , ... P ; = P ; ( P1 , ... Pn , 91 , . Qi = Qi ( P1 , . . . Pn , Y1 , ... In ) , Q1 , ... Qu ) , ( 3 ) P\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA116\u0026dq=dziobek+mathematical"},{"page_id":"PA120","page_number":"120","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. 16 . THE PARTIAL DIFFERENTIAL EQUATION OF HAMILTON AND JACOBI . In the integration of the systems ( 1 ) dp , OH дн dq - дн dp ; \u0026#39; ( i = 1 , 2 , ... n ) dt Əqi dt 2n constants , a1 , a , ... a ,, P1 , P2 , ... P are\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA120\u0026dq=dziobek+mathematical"},{"page_id":"PA122","page_number":"122","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. But the second member of this is = 0 , since it is the total deriv- a w , that is of a constant with respect to the time . ative of Hence даб ( 9 ) alat w + H :) 0 : да The equations ( 7 ) and ( 9 ) show that ( 8 ) is a\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA122\u0026dq=dziobek+mathematical"},{"page_id":"PA123","page_number":"123","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. with respect to a parameter a ,, it follows , since it is contained only in W , that ( aw да : a w a 0 and hence , by ( 13 ) , at д a ( aw ) да at + др at = да др እ д მ 1 ; ( 8 ) др or Hence , the expressions , ( 14 ) d a\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA123\u0026dq=dziobek+mathematical"},{"page_id":"PA126","page_number":"126","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Hence the substitution , which changes W into V , need not be taken into consideration , and we get the following final result : If there is a system of total differential equations ( 16 ) dp : dt дн dqi дн - = 9 dqi dt dpi\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA126\u0026dq=dziobek+mathematical"},{"page_id":"PA130","page_number":"130","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 2T - h ) dt = 2 Tdt - h ( t - t \u0026#39; ) = V - h ( t - t \u0026#39; ) . = √ ( 2r With the limitations now made , U + T ) dt = 8 V —s [ h ( t — t \u0026#39; ) ] • f ( U + r ) t / = Σ av av av -x : + dy ; + 87Zi θα dyi dzi av av dy ! sy ! + əzi\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA130\u0026dq=dziobek+mathematical"},{"page_id":"PA134","page_number":"134","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. 2 2 2 2 ( or ) 2 + ( or ) \u0026#39; + ( or ) \u0026#39; = ( or ) 2 + ( or ) \u0026quot; дх ду др др 2 ( r + r ) 2 +2 avav ( ( r + r2 ) _ 1 ( ~ + ~ ) 2 — p2 ) . Or Op Therefore the equation ( 4 ) takes the form ( 7 ) h = Əv Əv p2 — ( r + r2 ) 2- dr ap\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA134\u0026dq=dziobek+mathematical"},{"page_id":"PA136","page_number":"136","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Since it vanishes for x = x , y = y \u0026#39; , z = z \u0026#39; , it appears that ( 13 ) is the action of the system from the initial to the final configura- tions , and it is in the desired form . The time required by the planet to\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA136\u0026dq=dziobek+mathematical"},{"page_id":"PA139","page_number":"139","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. getting the notable equations ( 17 ) and ( 18 ) from those §1 . This example shows how a general theory may throw a new and sur- prising light on an old and special problem . Jacobi has shown that the partial differential\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA139\u0026dq=dziobek+mathematical"},{"page_id":"PA140","page_number":"140","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 20 ) 2n V - No1 S \u0026#39; . Hence it appears , that while the area described by the radius vector from the sun is proportional to the time the area described by the radius vector from the other focus is propor- tional to the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA140\u0026dq=dziobek+mathematical"},{"page_id":"PA141","page_number":"141","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. lytical mechanics with a new principle , —that of varying action . He introduced the idea of the action as the integral of the kinetic energy with respect to the time from one configuration to another , proposing it as a\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA141\u0026dq=dziobek+mathematical"},{"page_id":"PA142","page_number":"142","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. canonical system . The method which has been given , for changing a given system of constants into a canonical system , is due to Bour . Many \u003cb\u003emathematicians\u003c/b\u003e , by their labors , have perfected and generalized Jacobi\u0026#39;s ideas\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA142\u0026dq=dziobek+mathematical"},{"page_id":"PA148","page_number":"148","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Since the mass of the earth is somewhat more than eighty times that of the moon , the approximation on neglecting ò rises 2 1 from the order ( 40 ) 1 to the order 2 . 400 1 = 1 80 12,800,000 * In what follows , by the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA148\u0026dq=dziobek+mathematical"},{"page_id":"PA149","page_number":"149","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Series ( 1 ) converges so long as ( t - to ) does not exceed a cer- tain limiting value . This follows from Cauchy\u0026#39;s investigation on the convergence of those series which represent functions . defined by differential\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA149\u0026dq=dziobek+mathematical"},{"page_id":"PA158","page_number":"158","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. FOR OR да dt , dt , ... де The form under which the perturbations now appear are called . special , in so far as they are reckoned from a definite instant to . These special perturbations were included in the ideas of the\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA158\u0026dq=dziobek+mathematical"},{"page_id":"PA160","page_number":"160","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. turbations are to be used . We can , however , according to the rule for differentiation under the sign of integration , first inte- grate with respect to t and then differentiate with respect to the elements a , e\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA160\u0026dq=dziobek+mathematical"},{"page_id":"PA162","page_number":"162","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. doz ( 4 ) and dt dx = A + B dt dy dt \u0026#39; d2dz ( 5 ) dt2 d2x d\u0026#39;y dA dx = A + B + dB dy + dt2 х dt2 dt dt dt dt y dA dx dB dy = —A − Bμ + Αμ dt dt + Finally by substituting ( 2 ) and ( 5 ) in ( 1 ) , we get ( 6 ) and from ( 3 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA162\u0026dq=dziobek+mathematical"},{"page_id":"PA164","page_number":"164","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 16 ) OR W = a да -2n ford R dt , θε d2P μ . P + W . dt2 equation ( 15 ) becomes ( 17 ) This equation is of the same form as ( 1 ) and is to be inte- grated in the same manner . ( 18 ) Thus we get Wdt t p_ Jawa_z fywa P\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA164\u0026dq=dziobek+mathematical"},{"page_id":"PA170","page_number":"170","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 11 ) in which r1 = ( a + bcos V + ccos W ) — , 12 a = r2 + r122 , b = -2r1r2cos J , c = -2rr , sin2 J. If , as we shall assume , r , is always greater than r2 , or r always greater than r1 , then since a is greater than\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA170\u0026dq=dziobek+mathematical"},{"page_id":"PA172","page_number":"172","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 20 ) Z 3z2 8 Aλ ^^ = { } ( 11 ) — ~ ~ ~ , r2 ( 3 ^ — \u0026#39; ) + 3 ~ r‚2r22 [ 2 ( 5 ^ ) + ( 5 ^ ~ 2 ) ] B = 2 5z3 16 · · r ̧3r ̧3 [ 9 ( 7 ^ ~ 1 ) + ( 7 ^ −3 ) ] + ... 3z2 · B` = \u0026quot; ~ \u0026quot; , \u0026quot; 2 ( 3 ^ ) — 3 = r2r ,\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA172\u0026dq=dziobek+mathematical"},{"page_id":"PA177","page_number":"177","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. where a , 71 , 72 run through all positive numbers including 0 , which satisfy the relation ( 40 ) . If g is odd we can put g − 2a = 2x + 1 = 71 + 829 where x may have all values from 0 to 1 ( 9-1 ) . Therefore x = ( g\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA177\u0026dq=dziobek+mathematical"},{"page_id":"PA178","page_number":"178","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. number for which they are zero . If we represent the number of the latter by Tg , then . 8 , = ( Sg + Tg ) . 2 If g is odd , Tg = 0 , for 7 , and 1⁄2 can not be even at the same time as must be the case when 4 , 1 , 2 are\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA178\u0026dq=dziobek+mathematical"},{"page_id":"PA182","page_number":"182","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. in ascending powers of these quantities . If these are all intro- duced into the above product and this again in the term ( 46 ) , the development of the perturbing function finally receives the following form : ( 49 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA182\u0026dq=dziobek+mathematical"},{"page_id":"PA183","page_number":"183","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. where , in general , Ик except for k = 0 , for which From these we find 2k ( k2 + 2 ) 3 u = 1 . s 。= 1 , s2 = 12 , 890 , Sε = 444 , 81 = 2 , 83 = 30 , s = 222 , s1 = 858 . If Leverrier had selected the complete expeditious\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA183\u0026dq=dziobek+mathematical"},{"page_id":"PA184","page_number":"184","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. velocities . We have also the following six partial differential equations OF OF + OF OF OF OF = ayı 0 , + = 0 , dy2 JZ1 + öz 2 = 0 , Oz2 х1х2 OF OF OF OF X1 dyi Y 1 J x 1 +22 дх + X2 JY 2 · Y 2 DX 2 = = 0 , ( 51 ) OF OF OF\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA184\u0026dq=dziobek+mathematical"},{"page_id":"PA186","page_number":"186","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. The numerical computation of these coefficients can be per- formed in two ways . On the one hand , after Liouville , ( Sur le calcul des inégalités périodiques , Jour . de \u003cb\u003eMath\u003c/b\u003e . I , 1836 ) , we may expand the double\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA186\u0026dq=dziobek+mathematical"},{"page_id":"PA190","page_number":"190","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Since , by ( 2 ) , the coefficient in braces remains unchanged when i is exchanged with i , we must have ( 12 ) or , by ( 10 ) , ( 13 ) - ( ( 8—2 ) 1 ) = α ̧ α1⁄2 [ k ( s1 ) — ( s \u0026#39; — 1 ) —- ( s \u0026#39; + 1 ) ] , ( ( 8-2 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA190\u0026dq=dziobek+mathematical"},{"page_id":"PA201","page_number":"201","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ( 18 ) ( 3y1 ) = Σ [ ( K ; 1 ) sin ( 4 + 51 ) + e , ( K ? ) sin ( + 71 ) + e1 ( K ; 3 ) sin ( +23 , −-- ~ 1 ) + e ( K ) sin ( +2 ) + e ( K ) sin ( + 252—2 ) ] , ( 19 ) ( ôz1 ) = Σ [ ¿ , ( K , ® ) sin ( 4 + 51—21 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA201\u0026dq=dziobek+mathematical"},{"page_id":"PA202","page_number":"202","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. denoted by a ,, ... They give another term { 8x , to ox1 , as follows : { dx } = 3n1 2a дх дх ( a , ε ) , { 51 } t + ( α , εξαι } + ... OH 051 The constants can be taken entirely arbitrarily . Laplace puts 2a1 [ OR1 ] { 5 }\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA202\u0026dq=dziobek+mathematical"},{"page_id":"PA204","page_number":"204","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. depend on the coordinates and velocities in the manner given in § 2. Whatever instant selected , the same values for the elements would always result . The actual state of the case is , that we get different elements for\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA204\u0026dq=dziobek+mathematical"},{"page_id":"PA205","page_number":"205","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. The component velocities also contain the time as before , that is ( 2 ) ( 3 ) dx дх = = x \u0026#39; = n , Θε дх dt Ot From ( 1 ) and ( 2 ) it follows that dx da dx de 0 + + Ja dt de dt dy da dy de 0 = + + da dt de dt dz da dz de 0\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA205\u0026dq=dziobek+mathematical"},{"page_id":"PA218","page_number":"218","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. 0 = 41,1 1 ( [ 1 , 1 ] —91 ) + α2 , 1 [ 2,1 ] + α3,1 [ 3 , 1 ] + ... + an , 1 [ n , 1 ] , and likewise , ( 20 ) 0 = ,, [ 1,2 ] + a2 , 1 ( [ 2,2 ] —g1 ) + α ,, 1 [ 3,2 ] + ... 0 = a ,, [ 1,\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA218\u0026dq=dziobek+mathematical"},{"page_id":"PA229","page_number":"229","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. %% also proportional to them . On the other hand , they are of the second order with reference to K and K \u0026#39; and , as numerical com- putation shows , they are so small that they may be neglected . While these terms may be\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA229\u0026dq=dziobek+mathematical"},{"page_id":"PA233","page_number":"233","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. get the effects of the secular terms of the perturbing function , while the theory of absolute perturbations is applied only to its periodic terms . It is remakable that this combination of the two methods was used earlier\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA233\u0026dq=dziobek+mathematical"},{"page_id":"PA240","page_number":"240","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. regarded as secular - secular terms . Hence the equations and dK \u0026#39; dt - dK dt = 0 0 are not fulfilled and this is entirely correct . In fact , these two equations are not generally correct ; the invaria- ble plane should be\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA240\u0026dq=dziobek+mathematical"},{"page_id":"PA241","page_number":"241","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. introduced into the expressions for the coordinates , the follow- ing general schematic representation of the coordinates : ( 26 ) ( 27 ) x = ΣK cos L , y = ΣK sin L , z = ΣK\u0026#39;sin L \u0026#39; . The K\u0026#39;s and K\u0026#39;s are coefficients\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA241\u0026dq=dziobek+mathematical"},{"page_id":"PA250","page_number":"250","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. = dt a [ 5 ] ́da1_d [ a ] d ( a , ) dt = + dt dt d ( ar ) _d [ a ] + © ( a ; ) [ n ] + [ a ] [ n ] + d ( a1 ) [ n ] + ... @ [ 5 ] + Jo ( a ) d [ a ] , d ( a1 ) d [ e ] [ a ] dt [ e ] dt + + + @ [ 5 ] dt d ( a )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA250\u0026dq=dziobek+mathematical"},{"page_id":"PA251","page_number":"251","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ond member of the first equation ( 8 ) , § 28 , if the secular values are used for the elements . This amounts to the following : Let the perturbing function R1 be expressed in terms of the elements , or ( 4 ) R = R ( a1\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA251\u0026dq=dziobek+mathematical"},{"page_id":"PA258","page_number":"258","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Let ( 5 ) ( 6 ) 1 m2 ( √ ( x , — x2 ) 2 + ( y 1 — y2 ) 2 + ( 21 — 22 ) 2 1 -- √ ( x1 — x2 ) 2 + ( yı — Y2 ) 2 + ( ≈1 — Z2 ) 2 = X 1 X 2 + Y 1 Y 2 + 21 22 3 Σ ( Psin + cos ) , X1 X2 + Y1 Y2 + Z , Z2 = ( p sin + cos ¿ ) . The\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA258\u0026dq=dziobek+mathematical"},{"page_id":"PA262","page_number":"262","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. Therewith the proof is completed that da dt can contain no sec- ular term even when the second powers of the masses are taken into account , and we can now , with greater emphasis than in $ 30 , state the proposition\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA262\u0026dq=dziobek+mathematical"},{"page_id":"PA264","page_number":"264","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. they are not included in the purpose of this work which is devo- ted solely to the study of motions resulting from gravitation . Yet , with this limitation , the doctrine of the eternal stability of the solar system is very\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA264\u0026dq=dziobek+mathematical"},{"page_id":"PA265","page_number":"265","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. by means of certain circumstances , -the smallness of the per- turbing masses , and the smallness of the eccentricities and inclinations . If this represents the facts , is it because it expresses a \u003cb\u003emathematical\u003c/b\u003e law , and\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA265\u0026dq=dziobek+mathematical"},{"page_id":"PA272","page_number":"272","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. + tsΣΣmx [ ( a , ON \u0026#39; a ON dc ; OK G\u0026#39;sin ( H - H \u0026#39; ) an \u0026#39; an + ( b , c , b ; on ) kg\u0026#39;sin ( h — n \u0026#39; ) ] . Since [ 3 , 0 ] is constant , the terms multiplied by t must vanish and the above must reduce to the constant term\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA272\u0026dq=dziobek+mathematical"},{"page_id":"PA281","page_number":"281","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ually displaced it , though curiously enough , not in its purity , but wonderfully mixed with the first theory . It was seen that the secular terms must eventually cause great changes in the ele- ments , but instead of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA281\u0026dq=dziobek+mathematical"},{"page_id":"PA283","page_number":"283","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. ter accelerated . On the other hand , Lambert had found that in modern times the opposite is the case . Laplace saw that the explanation of this apparent contradiction of the law of the invariability of motions was to be\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA283\u0026dq=dziobek+mathematical"},{"page_id":"PA292","page_number":"292","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. The sixth table gives the course of the Earth\u0026#39;s elements for a period of 200,000 years . ELEMENTS OF THE ORBITS OF THE MAJOR PLANETS . I. Epoch . Planet . Mean Eccentric- Distance . ity . Mean Moon . P Paris . Mean\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA292\u0026dq=dziobek+mathematical"},{"page_id":"PA294","page_number":"294","snippet_text":"Otto \u003cb\u003eDziobek\u003c/b\u003e. V. 91 = = 0.692870 Y = 0 92 = 2.842232 72 = - 0.756015 93 = 3.780294 Y3 =3.106931 94 = 22.500087 - Y4 = — 25.952538 96 = 5.2989 Yo = =4,795350 . 965 = 7.5747 Y6 = 7.067951 97 = = 17.1527 Yy = -17.468102 9. = 17.8633 Y\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=O9RNAAAAMAAJ\u0026pg=PA294\u0026dq=dziobek+mathematical"}],"search_query_escaped":"dziobek mathematical"},{});</script></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>