62 A NEW METHOD OF DETERMINING 
Let v the angle between the positive axis of X and the radius-vector measured in 
the plane of the disturbed body, here called the plane of X Y. The differential coeffi- 
cient of the perturbing function © relative to the ordinate Z perpendicular to this 
plane is found by differentiating © relative to z and afterwards putting z = 0. 
Thus from 
m J rr’ i 
(Oa Le |; 
ae cle 
1tm 
AO = (Gaal Pe (Qa P se (ee) 
r - r? aoe ory’ #, 
we find 
dQ m [- Wy 4p &: | 
dv 1+ m 2) ky > ihe 
dQ mm [— 1 Gate A 
dr ~~ Vem E 4 ) Pl? 
mm 1 9 CB 
dQ = aoa [— = Ua =| 
dA dH dA dA Z 
== = Hf = A— = r—?H, == = — =, 
dv dv’ dr 2 dz 4 
Hence 
dQ m’ 1 1 ery 
du 1+m Ls a wr id 
dQ m’ il m’ 7 
TLidh € Raye (Asem = aller A yn 
GQ __ mm 1 1 a iapieg ae ois Ay j 
@ = tole ja |sin 2.7 sin (f’ + I’) 
where 
HY = sin (f +11) cos(f’ + TI’) — cos J cos (f+ UH) sin (f’ + I’), 
2 = —r’.sin Jsin (f’ +541’). 
