665] 
that is, 
and similarly 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
ЭA = q ЭП — %B du, 
185 
dB = ^dtl-%A du, 
whence 
А dB-BdA= — %(A 2 - В 2 ) du. 
Proceeding to a second differentiation, we find 
d 2 A = Э 2 П - ~ ЭП du + lA (du) 2 , 
d 2 B = ^ a 2 n - ^ an aw + ib (du) 2 , 
and thence 
A 
A d 2 A - (dA) 2 = ~ {П Э 2 П - (ЭП) 2 } + i(A 2 - B 2 ) (du) 2 , 
В d 2 B - (дВ) 2 = ^ {П Э 2 П - (ЭП) 2 } +i(B 2 - A 2 ) (du) 2 . 
To simplify these we assume (as the third equation above referred to) 
П Э 2 П — (ЭП) 2 = 0. 
The last-mentioned two equations then become 
A d 2 A - (dA) 2 = i(A 2 - B 2 ) (du) 2 , 
В d 2 B - (dB) 2 = l (B 2 - A 2 ) (du) 2 , 
which several equations contain the theory of the functions А, В, П: we have as 
their general integrals 
A = \Ae Xu \fa — b + e~^ {u+v) ], 
B = — %Ae Xu \/a-b {e* (w+v) — e~* (u+v) }, 
n = Ae Xu , 
where A, X, v are arbitrary constants. Forming the quotients A : n, B : fl, and 
introducing the notations cosh, sinh, of the hyperbolic sine and cosine, also writing 
for simplicity v=0, the equations give 
\f a — x— Va — 5 cosh w, 
= —\f a —b sinh | w, 
which express the integral of the differential relation 
dx 
du = 
Va — x .b — 
C. X. 
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