THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 75 
If we wish to use polar coérdinates we have 
d° = Roos f — 8 sin f 
daz : 
2 5 
= = Rsin f + S cos f, 
From 
=P COS ih, TST sm, 
we find 
dx = dr cos f — rdf sin f 
dy = dr sin f + rdf cos f 
a= dr cos f—rd7f sin f — 2dr df sin f — rdf? cos f 
@y = drsin f + rd°fcos f+ 2dr df cos f— rdf? sin f 
From the expressions for dx and dy we find 
dy cos f.—dxsin f= r df 
dx cos f.+ dysin f = dr, 
and hence 
da 1 Ge de 
— ™ —-— .—~ sm — GO 
ae peg hs te Oey 
do 1 ag ae 
—= 2,60 == ii 
dy pa COR ae a BID 
from which we see that 
R=VA+m) 2 saPa+m)i@ 
If we multiply the expression for d’w by cos f, that of dy by sin f, and add, 
we obtain 
ax cos if a @y sin f — Gp df. 
