THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 67 
‘When these are known we multiply the function ua’ (ey by 
r\2 ean sinI 7. , , sinir. . 
Le —i. eal =. osin(f +1), eT ain (f + I), 
(a cosl . 
a’ ? a b) 
: 5 
the function ua (=) by 
@ |r lr]? 3 sinlr’ . ; ar \ene Ig 
alae pale oa “sin (f+ WW) [7 — 2 a; 
aie J PR ay ae 3 sinl 7 7 il 
© = ope tp WO), 5 “sin( f+) [7 aa 
pe E stale a) em (Gg? Sone 
We will now find the expressions for (7), (7)’, (7)”, and for the various factors 
just given, that are the most convenient for numerical computation. 
We have 
(1) = “sin (2). sin (f +1’). 
Putting, for brevity, 
b = — *, cos ¢’sin J cos Il’ 
6.) = /sing/esinplil 
3 
and noting that 
a/\? sin f’ (0) QF. @- LOA. 
() ae ms = Fe + J, ] sin g’ + ole = The sin 2g’ + ete. 
(2) cos ie eee Tet | cos J! ++ a Fl cos 29’ + ete. 
