12 
[632 
632] 
then y is 
or, what i 
the consta 
that is, 
632. and the vj 
ON ARONHOLD’S INTEGRATION-FORMULA. 
[From the Messenger of Mathematics, vol. v. (1876), pp. 88—90.] 
where the 
The fundamental theorem in Aronhold’s Memoir, “ Ueber eine neue algebraische 
Behandlungsweise der Integrale...II (x, y)dx, &c.,” Grelle, t. LXI. (1863), pp. 95—145, is 
a theorem of indefinite integration. The form is 
A f dx = lo (ag + hy +g) x + (Ag + by +/) y + g% +f v + c 
J(ax + ßy + y) (hx + by +/) ° ax + ßy + y 
where y is a certain irrational function of x, determined by a quadric equation, and 
the other symbols denote constants connected by certain relations; viz. writing, for 
shortness, 
U = (a, h, c, f g, K§x, y, l) 2 , = (a,...\x, y, l) 2 for shortness, 
that is, 
= ax 2 + 2hxy + by 2 + 2fy + 2gx + c; 
W=(a, b, c,f, g, Kfix, y, l$g, y, 1), = (a,...\x, y, l$g, y, 1), 
that is, 
= (ax + hy+g)% + (hx + by +f) y+gx +fy + c, 
or 
(a£+ hy + g) x + (Ag + by +/) y +g% +fy + c; 
(P , Q , R) = (ax+hy+g, hx + by +/, gx +fy + c), 
(P 0 , Q 0 , R 0 ) = (a^+hy+g, Ag + by +fi g^+fy+c), 
il = ax + /3y + y, 
= ag + fiy + y, 
(A, B, C, F, G, H) = (be -f 2 , ca - g 2 , ab - h 2 , gh - af hf- bg, fg - ch), 
The v 
when 
P dx 
Q 
dx, 
, tb 
that is, sul 
On the rig 
(as at once 
The equati