280 A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. [212 
where on the left-hand side il = il (r, ]>, o, d, <f>). The function il so expressed 
satisfies, of course, the partial differential equation 
dO dO _ 
d]> do 
(which conversely implies that p>, o only enters through the function p - o), and it 
also satisfies the partial differential equation obtained from the before-mentioned equation 
dO ,dO ,dO 
cot z ~rr — cot 9 cosec <p 
dcf) dz 
dd 
(îî = fi (r, z, e, 
by the introduction of the transformed expressions of the differential coefficients, and 
which may be written 
riil , , dO dO 
cot z -j-r — — cot <p —j cosec <f> 
d$> dcr 
dd’ 
where il = il (r, |?, cr, d, <f>). 
Using the last-mentioned equation to transform the value of dcf), the expressions 
for the variations become 
dr = 0, 
d]> = 0, 
d 
nae sin f _ dO 
Vl - e- ~ dr 
dt, 
dna 2 V1 — e 2 = ^ dt, 
dcf) = 
— cot (f) 
dO 
na 2 V1 — e 2 
do 
cot cf) 
dO 
na 2 Vl — e 2 
dcf) 
cosec cf> 
dO 
no? V1 — e- 
dcf) 
rf ,__cosec0 d_n dt 
net 
;2Vl-e 2 dd 
dt, 
where 0 = 0 (r, |>, cr, d, cf>) as before. 
I suppose now that the orbit of the planet, instead of being referred to a fixed 
plane, is referred to a moveable plane or orbit of reference. It is assumed that the 
longitudes in the orbit of reference are measured from a departure-point defined as 
above,—that is, from the point in which the orbit of reference is intersected by any 
orthogonal trajectory of the successive positions of the orbit of reference. And the