THE OKI] IT OF U II AX US. 97 



where 



^^ = (2". 31 — l".-247) 7^+ 0".13r» 



Putting in the above expressions 



o = 95^3' + 3459" r, 

 cosu = —.0880 — .01677; 

 sin <j = + .9961 — .00157; 

 we find 



(ft.c.O) = — ((V'.ll — 0".06 07' 

 (b.s.l) = — (0 .-20 — .ll//)^— 0".05 7'» 

 (5.c.l)= (2.30 — 1 .24u) 7^+0.127'* 

 (6.s.2)= —0.017" 



(&.c.2)= (O.U — 0.06;/)r. 



We have, finally, to consider the terms of long period in hy; and <k which have 

 been omitted from the periodic perturbations produced by Neptune, in computing 

 the terms of bi3 on page 61, and which are as follows: 



hy; = 1".43 cos (2/' — g) — 0".39 sin (2/'— .7) 



— 2.12 cos {ir — 2<j) + 1 .00 sin (4/'— 2</) 

 + .20 cos (6r — 3^) — .04 sin (61'— 3^/) 

 + constant = 0".00 



^k= 0".80cos(2Z'— f^) — 2".2Ssin(2/'— <j) 



— 1 .06 cos (4Z'— 2-7) — 1 .85 sin (4Z'— 2y) 

 + .04 cos (6Z' — 3r/) + .19 sin (6l'—S<j) 

 -f constant = 0".364. 



For the period during which Uranus has been observed, these values of Sy; and hk 

 may be replaced by the following: 



^y,= — 0".80T 



a- = + 0.277 



which are to be multiplied by the factor 1 -\- ^i. The corresponding perturbation 



of the latitude will be 



^^ = sin v^.y; — cos \''k. 



Putting for v its approximate value 



\ = r/ -{- a -\- 2e sin ff 



and developing to quantities of the first order with respect to the eccentricities, 



we have 



sin V = sin (g + u) -|- e sin (2^ -{- u) — e sin « 



cos V = cos {(J -|- (o) + e cos {Qg -{-u) —e cos cj. 



13 May, 1873. 



