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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