665] 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
193 
equations, which for an arbitrary value of M would be inconsistent with each other : 
it is therefore necessary to verify that the value assigned to M is such as to render 
the three equations consistent with each other, and this will accordingly be done. 
The foregoing equation 
ii 0 2 i2 - (diiy = №M, 
where M has its proper value, (or say the three partial differential equations into which 
this breaks up), constitutes the other equation above referred to, which with the original 
equations A = ft Va, etc., serve to define the sixteen ^--functions and ÎÎ. 
The remainder of the present memoir is occupied with the analytical investigation 
of the foregoing theorems. Although the mere algebraical work is very long, yet it 
appears to me interesting, and I have thought it best to give it in detail. 
The equations 
The analytical theory : various subheadings. 
du=% + 
V Л 
dy 
VF’ 
dv 
give 
Qdx ^ „ 
= ov-ydu, 
хдх уду 
VX + VF’ 
— = dv — x du, 
VI 
which determine dx, dy in terms of du, dv. A different form is sometimes convenient; 
writing dur = dv — a du, and recollecting that a, a! denote a — x, a — y respectively, the 
equations become 
в dx _ a 
—— = dvr + aj du, 
V Л 
Ody 
VF 
= Эта- + a du. 
Expression for d fa. 
We have 
dfa = d Vaaj =—(ad^ + Эа) = —= (a dy + a x dx) 
2 Vaaj 2 vaax 
= —\{afY(dv — x du) — ^ \ r X (dv - ydu)}; 
2 Vaa x v 
substituting for fX, fY their values Vabcdef, Va 1 b 1 c 1 d 1 e 1 f 1 , this is 
dfa = \ {VabAdAf, (dv - x du) - V a^cdef (dv - у Эм)}, 
and by the mere interchange of letters we can of course find Э fb, etc. 
C. X. 
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