( 1243 ) 
y OF, Ch” Or, — 
Oy Or 7 
OF, OF, ; 
a v ee nk, 
we obtain 
(eX + yV)(@Y — yX) Pr_s — U(a? Hy?) + n (aX +yY)F,=0 
which proves that U is divisible by «NV + yV 
If therefore 
U = (aX + yY) Pis 
we have 
(@Y — yX) Pos — (@? + YI Bres Fr nk = 0 on ees Ne 
and the conditions for a centrum may be written 
dBi . OF ashe 
ae y = 2(#X + yV) 
OF, OF, OF, ORs e SN 
av dy mt Ul Fr Ef He fe ) Oy = (7X + u} ) ie 
Oy Uke de, OE 
v Oy y de EE X De + } dy — (aX + y ) ) Pe, 
where evidently /; represents a homogeneous function of order 7. 
These conditions may be further reduced, for 
OF 42 > OF 49 
X + jee == (wX -+ yy) En 
Ox Oy 
OF ie OF 4-2 5 7 
r a ~- y en == (@X +4) Pii 
y | a 
give 
OF, 19 
re =uwP, — Pint 
Ox 
OF 49 
sil =e, af eS + yPy 
dy 
henee 
0 : 7 
oe wP, ae } Pad en XP, 1 + P| 
Oy : | 
or 
Remarking that P,= 2, the conditions (2) may finally be replaced 
by these 
