46 
Mathematics. — “New researches upon the centra of the integrals 
which satisfy differential equations of the first order and the 
Jjirst degree.” (Second Part). By Prof. W. Kaprnyn. 
8. Assuming in the third place 
a’ + ¢ =i(a+oc) 
aa’ — ec = (b—ib') (ae) 
2b’ = 8a + be 
or putting 6= 7p 
2a' = — i (8a—28-+ 3e) 
2e = 1 (5a—28 + 5e) 
2b' — 3at be. 
We have 
q =a — i (Ba+2b!) = — — (15a—28-+ 180) 
: 1 
9, = 2a 4 3d'—ib = — (180-4284 150) 
ete 5 (86a" 4+ 26a8 + 179ac — 48? + 28Be + 99c2) 
1 , 
== (45a? — 36ag + 84ac + 428? — 32e + 39°) 
ro = (1 800" — GaP + 265ac — 48? —8Be +137”) 
1 
i a (421la® + 11643 + 972ac — 128? + 120R¢ + 567c*) 
and for the coefficients of P, 
s, = (5a4 26) r, + a'7, 
2s, — 4s, = (8b+4 2c’) r, + (dat 40') r, + 2a'r, 
as, — ds, = der, + (Ob HC) 7, + (Bad 6b') r, H3a' r, 
As, — 2s, = 2cr, + (4bH40) r, + (2a+ 80’) 2, 
— s, = Cr, + (2b+5¢) r,. 
To determine the next condition we introduce the two following 
polynomia 
Pete, + tary + tag + tay? + tag + toy? 
P, =u ar + ue ry + uty? -+ u,v*y®? Hur yt + u,ay'+u,y°. 
The coefficients of the first are determined by the relations 
