82 
PROFESSOR DONKIN ON THE 
according as h, b are or are not a conjugate pair, i. e. of the form a j} bj. (The value 
+ 1 belongs to h = a j , b—bj, and — 1 to the converse.) 
According to the notation proposed at the beginning of this paper, the above 
formula may be written 
4M)__ %*^)_, 
*' Xj) *' d(h, k) — 5 
= 0. 
By a usual and convenient abbreviation, the sum 
2 d{h, k) 
; d iji, *i) 
may be denoted by the symbol* [A, &]. We have then, by (22.), 
[a ( , bj = - [b a aj — 1 [«;, bj] = [a i} aj] = [b it bj\ =0, (23.) 
j being different from i ; and, obviously, 
0„ aj = [bi, ftj=0. 
Now let f, g be any two functions whatever of the 2 n constants a x , &c. b x , &c. ; when 
the latter are expressed in terms of the variables ( Hyp . II.), f, g become also functions 
of the variables ; and if h, b represent, as above, any pair whatever of a x , &c., b x , &c., 
we have (see art. 1.) 
d{f,9) _^W,g) d{h, k) ~) 
x i) yd(h, k)‘d{y h x t )j 3 
the summation referring to h, b, and extending to every binary combination. 
If, now, we sum each side of this equation with respect to i, we obtain 
£/■.*]=*{[*. *]•! («•) 
(the summation referring as before to h, b). But, by (23.), \h, ti] is 0 unless h, b be 
a conjugate pair, and then it is +1 ; so that (24.) becomes simply 
d(f,ff) 
(25.) 
an equation which, written at length in the common notation, is 
2 / df_ dg_ _ (M. ! l l_ 3 L 
1 \dy t dx t dxi dyi) i \da i db t dbi dai ) ' 
The expression on the right being a function of the constants a x , &c., b x , &c. only, the 
equation (25.) expresses obviously the following theorem. 
If f=<P(x x , ... x n , y x , y 2 , . . . y n , t) 
g=jy(x x , x 2 , ... x n , y x , y 2 , ... y n , t) 
be any two integrals of the system of simultaneous equations (16.), (17.)? art. 6, then 
* Poisson employs the notation ( h , k), which would have led to confusion if adopted here. Lagrange (in 
the Mec. Anal.) uses [ h , A], but with a different signification. See below, note to art. 34. 
