42 
MR. LUBBOCK’S RESEARCHES 
of Euler is not so simple, but is remarkable as introducing the employment of 
three rectangular coordinates and the decomposition of forces in the direction 
of three rectangular axes. 
Although D’Alembert and Clairaut made use of the same differential equa- 
tions, disguised under a different notation *, yet they did not arrive at these in 
the same manner, nor did they employ the same method of integration. 
Laplace has pushed the approximations to a much greater extent ; but his 
method coincides in all respects with that of Clairaut. 
In the method of Clairaut, when the square of the disturbing force and the 
squares of the eccentricity and inclination are neglected, the equations em- 
ployed are 
r f /d R\ 1 /AR\Ar\ n 
-Yd r \Tr)-T{in.)r>.! = 0 
A R 3 m, r- . , 0 , 0 . N 
dX = ^r s,n(2A — 2X '> 
= -lirl{ 1 + 3cos (2*-2*,>} 
A.R 
dr 
h 2 = pa 
A 2 — 
1 + e cos (c A — ■m) 
dr • / , * 
— = c e sin (c A — vr) 
d A 
r 1 1 3 m. /*r 4 . , 0 . , , , 
-rrr H — / — sin (2 A — 2 A.) d A 
o A- r a pa‘ l J r 3 v " 
+ 
p- — ( 1 + 3 cos (2 A - 2 A.) ] 
2 p a r 3 I V p J 
3 m t e r 2 
‘fpr 3 
sin (2 A — 2 A,) sin (A — -zzr) = 0 
In order to integrate this equation, the value of in terms of \ must be sub- 
stituted, which substitution is an operation by no means simple, and therefore 
liable to occasion error. 
* The neglect by mathematicians of care in the selection of algebraical symbols is much to be re- 
gretted. “ La clarttS des iddes augmente h mesure que Ton perfectionne les signes qui servent a les 
exprimer.” 
