200 
MR. IVORY ON THE THEORY 
The new partial differentials of R represent the disturbing forces reduced to 
new directions. By combining the formulas (B), we get 
d R _ d R cos x R sinX rfR s 
dr ~~ d x • y' i + s 3 + dy ' */ T+7 3 ' dz ’ y' ] + s * ’ 
R d R d R 
and it will readily appear that the coefficients of are the re¬ 
spective cosines of the angles which the directions of the forces make with r ; 
so that is the sum of the three partial forces that urge the planet from the 
sun. In like manner it may be proved that ^ ^ + s - is the disturbing 
force perpendicular to the plane passing through the sun and the coordinate z, 
d R 1 -f. 5 2 
that is, to the circle of latitude; and that -jj . —-— is the force acting in the 
same plane perpendicular to r, and tending to increase the latitude. 
2. If the equations (A), after being multiplied by 2 dx, 2 dy, 2dz, be 
added together, and then integrated, we shall get this well-known result. 
^ + + ~=2/VR, 
ft. dtr r ' a J ’ 
( 1 ) 
in which — is the arbitrary constant, and the symbol d! R is put for 
rfR 
dx 
dx + 
d R 
dy 
dy + 
d R 
dz 
d z ; 
that is, for the differential of R, on the supposition that x, y, z, the coordi¬ 
nates of the disturbed planet, are alone variable. If we conceive that R is 
transformed into a function of the other quantities r, X, s, we shall therefore 
have 
„ d R 7 ,JR, , d R 7 
d’ R = ^7 dr + ^ nr dX + —ds. 
d\ 
ds 
J 
Supposing that the radius vector r, at the end of the small interval of time 
d t , becomes equal to r + dr, and that dv expresses the small angle contained 
between r and r + dr, we shall have 
d r 2 r 2 d v 2 —d x 2 + d y 2 + d z 2 ; 
for each of these quantities is equal to the square of the small portion of its 
