665] 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
187 
etc.; and, without in any wise fixing the value of il, we in fact find that each of 
these expressions is a sum of products; the form is, as will appear, 
A dB — B BA = ail 2 Vc — x .d — x — v CD, etc.* 
Passing to the second derived functions and forming the combinations A d 2 A — (dA)- T 
etc., each of these will contain a multiple of il 0 2 il — (0il) 2 , but if we assume this 
expression il 0 2 il — (0il) 2 = fl 2 M, where M is = (du) 2 multiplied by a properly determined 
function of x, then it is found that each of the expressions in question Ad 2 A — (dA) 2 , etc., 
becomes equal to a sum of squares, that is, to a linear function il 2 (X + fj,x): viz. it 
is equal to a sum of squares formed with the squares A 2 , B 2 , C 2 , D 2 . 
The foregoing equation 
il 0 2 il — (0il) 2 = Cl 2 M, 
where M has its proper value, is the other equation above referred to, which, with 
the equations A=£l\/a — x, etc., serves for the definition of the functions A, B, C, D, il; 
it may be mentioned at once that the proper value is 
M — |(0tt) 2 [-2x 2 + x(a + b + c + d) +«], 
where k is a constant, symmetrical as regards a, b, c, d, which may be taken = 0, 
but which is better put 
= a 2 + b 2 + c 2 + d 2 — ab — ac — ad — be — bd — cd. 
For the proof of the formula, I introduce and shall in general employ the 
abbreviations (a, b, c, d) to denote the differences a —x, b — x, c-x, d — x: the 
differential relation between x, u thus becomes dx = du Vabcd. I use also the ab 
breviations il 0 2 il — (0il) 2 = Ail, etc. 
We have 
AdB-BdA = ïl 2 (Va0 Vb - Vb 0 Va), 
the terms in 0il disappearing: viz. observing that da = dh = - dx, this is 
or observing that a —b — a — b, and writing for dx its value = Vabcd du, this is 
AdB-BdA =-\(a-b) il 2 V cd du, 
= — ^ (a —b)£l 2 *Jc—x.ci — x du, 
which is the equation expressing AdB-BdA as a sum of products: it is further 
obvious that the value is 
= — \ (a — 6) CD du. 
* It is hardly necessary to remark that a, v contain each of them the factor du; and the like in other 
cases. 
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