4i 



T n E ORBIT OF URANUS. 



The vulucs of ' ' C I)'iR„<It are formed from - ,' D\R by simple multiplication by 

 'Hi *y lit H 



V, and proper changes of sign. The values of /.-, and Z-. are then formed by adding 

 the terms of 2 Jj D\Ii^dt to the corresponding terms of — ^• 



Perhirhailons of radius vector. 

 Let us now resume equation (19), and put for brevity 



J/= —y- r. (44) 



If we give to w the successive values 0, -fl, —1, +2, —2, +3, —3, we have 



+ i 2, (i>i'/(.-i) +i'(.-i)7.) (''.— J'(i-o) < ^^"c cos (iV^+ <7) +Z-, sin(A''+jjr) j 

 r,%=i/X <! +i2,(i>i7i^+„+i>(Hi)7.)(''.— >'-(>+i))!^'eCos(i\^— (/)+A:,sin(iV— <7); ■ 

 + 1 Mlh%-^) — P(-2)'7.) («'— 'V->)) 1 ^'-ocos (iV^+2i/)+A-,sin(A^+2y) j 

 -|- etc. etc. etc. etc. 



the finite integral being taken with respect to all values of i from — oc to -|- oc, 

 and the terms in which the angles N^ug vanish being omitted. Proceeding far-, 

 thcr to expand with respect to i, if we collect similar terms we shall find the indi^ 

 vidual terms in ?-\^p to be as follows: 



^ i + IhQ^ (''2 — '-2) 



r poQi (''1 — ivi) 



+ ^ ] +(M3+M2)03-v-.) 

 I -\- etc. etc. 



f PoQi (''0 — I'-i) 



I 1 j^ I + {P'l2~{-p,qi) ('■! — 1V2) 



I + (Ms +J"372) (''2 — ^'-3) 



I -|- etc. etc. 



f Poqi (i'2 — j'o) 



(. -{- etc. etc. 



r Poq^ (''0 — i'-2) 



+ 1 ili j + (2^,*73 +i'3?i) (ri - r_,) 

 {. -\- etc. etc. 



r i'o^s ("3— j'o) 



* 1 + (i'l72— Ps^l) (^'2 — J'l) 



-j- etc. etc. 



i 1 7i-^ cos iV -f /'■« sin iV | 



)^ I /.•„ cos {N-\- g) ^h sin (i^+ g) 



\ 1c, cos (iV^ - r?) + /.:, sin {N-g)\ 



[1; cos (iSr+ 2i/) -f- /., sin {N-\- 2g) | 

 I /.-, cos {N- 2g) + 7.:, sin (N - 2g) \ 



- I Jc, cos (i\r+ Sg) + /.-, sin (i\^ + Sg) | 



