Variable Elements of a Disturbed Planet, 359 



And if in x'^+y'^ + z'^=^ + r\^=r'^ we substitute for a:, 5^, 

 and z their values from (1.), it will become identical. Equal- 

 ing therefore the coefficients of ^2, ^>j and ij^ to nothing in the 

 result, we have 

 A2+C2+E2«i, B-2+D2+F2=l, AB+CD + EF=o. (3.) 



These may be proved by putting for A, B, C, &c. their 

 values given above. 



Differentiate the two first of (3.), and 



ArfA + CrfC + EcZE=0, BrfB+DrfD + FtiF=o. 



These by (2.) will give 



(tan i sin 5 A — tan z cos 9 C + E) c? E = 0, 

 (tan i sin 9 B — tan / cos 9 D + F) ^ F = 0. 



Divide the first of these by c?E, then multiply by c?F; di- 

 vide the second by rfF, then multiply by </E, the results by 

 (2.) will be 



ArfB + C^D + ErfF = 0, Bd;A + D^C + FrfE = o,l 

 A^A + CrfC + E^E = 0, Bt^B + D^D4-F^F = 0,J' ^ ' 

 the two last being added from above. 



Since 



i£ - \^ A. nil 

 dt ^ dt'^ dt' 



d denoting the variation of i only. Therefore 



^=A?^+B^ + SA^^ + 8B^l 

 dt dt dt dt dt 



And A.r = AA^+ BA») + ^AA + )3AB. 



Multiply the two last ; there results 



+ B8B A,^-] + K^Ki.§ + BAAf^. AASA^I 



H-AA8BJ^^+AAB,^^BAB,^ + ABSA/5| 



+ AB8B,^^. 

 In this result change A into C, B into D, and we have the 

 expression of At/ -~. Again, change A into E, B into F, 



