the Motion of a Disturbed Planet. c 2$ 



sins" sin .&=sirw sin S + / (coszsindrfz 4- sin/cos^rf-&). 



Substituting these values, and changing u into A, and put- 

 ting it under the sign of integration, we obtain 



sin <f> ss sin i sin (u — S ) + I sin (A — d) cos i d i 



— / cos (A — -5) sin i d •&. 

 ~ ,. dt dR cosidtdR 



But dt — r—- : ~TTi «•& = j—. -— r-r. 



A sin 2 a # sin i d i 



These values being put in the above, it will become 



. . . . . . , /*cosidt dR . , 



sin$ =sm * sin (« - S ) +J j^j ~jj sin (A - S) 



/cos idt dR . RN 

 ___ cos (*-*). 



To abridge, we may write this 



sin <f> = sin i sin (u — $ ) + Q» 

 or sin <p = sin i sin (o + y/ — <& )— y /sin /cos (u + 7/ — 3 ) +Q. 

 Here y t is the regression of the node, and 7 is to be so de- 

 termined as to take away from Q the terms having t in their 

 coefficients. 



If in the equations 



sin <$> = sin i sin (u — • 3), 



dsin 6 . . , ., c?w 

 — ; — I = sin z cos (u — 3) -7-, 

 dt dt 



we change $, w, 1, and 3 into <p + 8 <$> u o + ^ °> *o + & *» anc ^ 

 •&o + 1*^ * $> & c « being the parts depending on the disturbing 

 force; and if we expand, taking account of the first power 

 only of 8 <p, &c, we shall find equations of the form 



8* = A8<f> + B8u, 



8d=C84> + D8y. 



From these we may correct the values of i and .& or 6 by 

 means of the corrections of <p and due to the disturbing 

 force, and in doing so we may take account of higher powers 

 of 8$, &c. 



I think the development of the preceding equations would 

 be attended with much less difficulty and perplexity than the 

 development of M. Hansen's. I have not noticed the reduc- 

 tion to a fixed plane, but must refer for that to the Number of 

 this Magazine for February 1844, where I have given equa- 

 tions particularly adapted to the lunar theory, and leading to 

 results expressed in terms of the true elliptic longitude. 



