142 
REPRODUCTION OF EULER’S MEMOIR OF 1758 
[404 
the numerator is 
— 2)s 2 dP V (G) + Fhs dTJ\J (G) sin U — Fhs V (G) cos U, 
and the denominator 
h~F' 2 + Gs~ — 22) Fhs sin U — /¿V sin 2 U, 
which, observing that h 2 = G — 2) 2 , is equal to 
(Fh — 2)s sin U) 2 + Gs 2 cos 2 U, 
and we have 
the integral of which is 
7| — 2)s 2 di7’+ Fhs sin UdU — Fhds cos U . 
# = /wT—ma , fr V (£) 
cf) + $ = tan' 
(Fh — 2)s sin ZPf + Gs z cos 2 U 
j Fh — 2)s sin U 
s cos Ü V (G) ’ 
where $ is the constant of integration, or substituting for h, s their values, the 
equation is 
F V (G - 3> 2 ) - D sin U</(K- ZLMNGu) 
tan (<f> + g) = ' 
It may be added that 
cos U V {G (K - 2LMNGu)) 
o) cos j = v — q + V {(^ - 3) 2 ) (K — 2LMNGu)} sin U], 
and therefore 
cosy = 
• _ ^ + V {(# ~ ^~ J ) ~ 2LMmu)} sin U 
G V (E -2LMFu) 
Euler remarks that the complexity of the solution owing to the circumstance that 
the fixed point P is left arbitrary ; and that the formulæ may be simplified by taking 
this point so that G — 2) 2 =0, and he gives the far more simple formulæ corresponding 
to this assumption ; this is in fact taking the point P in the direction of the normal 
to the invariable plane, and the resulting formulæ are identical with the ordinary 
formulæ for the solution of the problem. The term invariable plane is not used by 
Euler, and seems to have first occurred in Lagrange’s “ Essai sur le problème de trois 
corps,” Prix de VAcad, de Berlin, t. ix., 1772. 
To prove the before-mentioned equation for d(f) ; starting from the equations 
. v 
cosy = cos a cos l + cos /3 cos m -1- cos 7 cos n = —, 
sin y cos cf> = cos a sin l cos \ + cos /3 sin m cos p + cos 7 sin n cos v 
sin y sin (f) = cos a sin l cos X, + cos /3 sin m sin p + sin 7 sin n sin V, 
we have 
cosy dj cos <f> — sin y sin <f> d(f) 
= — sin a da sin l cos X — &c. + cos a cos X cos Idl + &c. — cos a sin l sin X dX + &c.,