158 
Proceedings of Royal Society of Edinburgh . [sess. 
s = A x + ax 2 + j3xy + yy 2 + . . . 
t = A y + ax 2 + p’xy + yy 2 + . . . 
u = A z + ax 2 + fi'xy -f y'y 2 + . . . 
where the values of s , A , a , . . . , t , A , a, ... u, A , a", . . . 
are sufficiently suggested* by giving one of them, viz., 
A =(a Vc) • 
and then seeks to find any function of the roots, F(^, y , z) say, in 
the form of a series proceeding according to powers of the constants 
s , t , u , the result being that the coefficient of s p t q u r in the said 
series is shown to he in general 
'Fix, y, z) V 
X p Y q Z r 
x 1 y 1 z 1 
where X , Y , Z are the variable members of the derived set of 
equations, and y is the determinant of their partial differential co- 
efficients with respect to x, y, z. 
Xo other case is dealt with, but the paper closes with the 
sentence — 
u Quae autem hactenus de duabus, tribus aequationihus 
inter duas, tres variabiles propositis protulimus, eadem 
facilitate ad numerum quemlibet aequationum et variabilium 
extenduntur.” 
Jacobi (1832, 1833). 
[De transformatione et determinatione integralium duplicium 
commentatio tertia. Crelle’s Journ., x. pp. 101-128.] 
[De binis quibuslibet functionibus homogeneis secundi 
ordinis per substitutiones lineares in alias binas transformandis, 
quae solis quadratis variabilium constant : una cum variis 
theorematis de transformatione et determinatione integralium 
multi plicium. Crelle’s Journ., xii. pp. 1-69.] 
The latter of these memoirs is by far the more important ; in fact, 
* In later notation the derived equations would of course be written — 
<r b c 
II 
a b c 
x + 
d b c 
r V c! 
a' b' c' 
d! V c' 
u b" c" 
a" b" c" 
d" b" c" 
x 2 +. . . 
