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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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