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 
J R d R cos X JR sin A J R s 
dr ~~ dx * V 1 + s s ' dy ' a/ \ + s 2 ~*~ dz ' y' 1 + s 2 ’ 
d R JR JR 
x ’ dy ’ dz 
are the re- 
and it will readily appear that the coefficients of -y 
spective cosines of the angles which the directions of the forces make with r ; 
so that -^7 is the sum of the three partial forces that urge the planet from the 
sun. In like manner it may be proved that ^ ^ + - - is the disturbing 
force perpendicular to the plane passing through the sun and the coordinate z, 
d R, 1 -f* 
that is, to the circle of latitude ; and that . — - — 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, 2 dz, be 
added together, and then integrated, we shall get this well-known result. 
J x~ + dy 2 + dz~ 
fj. . df 2 
V = 2 / rf ' R . 
( 1 ) 
in which — is the arbitrary constant, and the symbol d! R is put for 
J R 
dx 
dx + 
% d y + 
JR . 
iu dz ’ 
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, \ s, we shall therefore 
have 
..-P, JR, JR, JR, 
d R — -^r d r + dX + ~rr d s. 
dx 
ds 
I 
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 _j_ r 2 d v 2 = dx 2 + dy 2 + dz 2 ; 
for each of these quantities is equal to the square of the small portion of its 
