( 1250 ) 
For small values of w and y the general integral may be expanded 
in the form 
Jy +S P+. = Const. 
which proves that in this case the origin is a centrum. 
8. If we assume in the third place 
a’ =b and 25’ = 3a + 5c 
the corresponding differential equation takes the form 
dy —a-+ ba? + (8a + 5e) zy — by —#+4+Y 
da yaar? + 2bay + cy? twee: 
Here j 
DX 07 
DE oe indd tee 
and omitting constant factors we have, as before 
ee 
P=y 
P, = bay + (4a + 5e) y? 
Pi “ (a + 2v) 0? + Baty + b (8a + 9) ay? + 
1 
+ (da? + 11500 + 367 + 75e)" 
ZKT » md 12 4,3 : 3 4 
P, =s,e* + say + 8,0%y* + SUY + 8,y 
where 
b 
e= —g [Ba* + 10ae — 35° + 100] 
1 
s, = 126’ (a + c) 
s, — b [44da* + 107ac + 3b? + 66c*] 
1 
8, = | [B08a? + 1245a%e + 57ab* + 1675a0* + 699]. 
Proceeding to 
Pit v' + taty + t,a°y? + te?y®? + t,xy* 4- ty’ 
and 
P,==u,2" Jury. + UY, 
we obtain 
t, = (9a + 5e)s, + bs, 
2t, — 5t, = 8bs, + (lla + 10c)s, + 26s, 
3t, — 4t, — des, + obs, + (18a + 15e) s, + 36s, 
4t, — 3t, = Bes, + 2bs, + (15a + 20e) s, + 4bs, 
dt, — 2t, = Zes, — bs, + (17a 4- 25a) s, 
—t, = cs, — dbs, 
——— 
