212 REPORT—1862. 
fanction, viz., assuming that there is a foree-function U independent of the 
dé de’ de ~ dy’ 
aa ) all that is required is to find a function © of w, y, z containing three 
arbitrary constants A, B, C (distinct from the constant attached to @ by mere 
addition) satisfying the differential equation 
2) +(8) (2) a0 
for then the required integrals of the ais of motion are 
dO_ 4, 40 ite ! 
TRA Spee GG=e+ 
A!, B!, ©! being new arbitrary constants. Liouville introduces in place of 
x, y, z, the elliptic coordinates p, yw, v, which are such that 
x y z 
ele + i. 
2 2 Fe 2 
ay y i alin 
2 2B e he pear d 
eo 7 y° 2 1 
YBa er 
or, what is the same thing, 
wall, 
ia VERE VENT 
Ve —B 
Pers VP —e EV e— 0 ae 
oN e—b? 
and he then finds that the resulting partial differential equation in p, p, 
may be integrated provided that U is of the form 
pa) fe +(p°— v7) Fut (p* =v av 
(e°—p*) (p?—v*) (u’—y*) 
f, F,; w being any functional symbols whatever: viz., the expression for 
Q is 
2 +A+B 2+ 2Cp* 
e=ld Wee Pp lv z ls 
-\% Be)” 
+4 area Fu + A+ Bu? + 2Cp 
(WH P)(C— eh)” 
+a is es Qa + A+B? +207 
(6°—»’) (¢’—yv’) 
In the case where U=0 we have a particle not acted on by any forces, and 
the path is of course a straight line. The peculiar form in which these equa~ 
tions are obtained leads to very interesting results in regard to the theory of 
Abelian integrals, and to that of the geodesic lines of an ellipsoid. 
The formule require to be modified in certain cases, such as c=6 or 6=0. 
The case 6=0 leads to the theory developed in the first memoir in relation to 
