560 A MEMOIR ON THE SINGLE AND DOUBLE THETA-EUNCTIONS. [704
138. Suppose that in this equation u' becomes indefinitely small. If u’ were = 0,
the values of X', Y', Z', W would be a, 0, 7, 0 : hence v! being indefinitely small,
we take them to be a, 3/3, y, 38, where
= (“' iu + v ’ i) Y > and dS ’ = (“' Tu + V ‘ il) w ■ (w = v = °>’
are, in fact, linear functions of u' and v.
We have if 4 if 0 — if 0 if 4 standing for
if 4 (u + u) if 0 (u — «') — if 0 ( u + u') if 4 (u — u),
and here
if 4 (u ± u!) = if 4 + 3if 4 , if 0 (u +11) = if 0 ± 3if 0 ;
the function in question is thus
(if 4 + 3if 4 ) (if 0 - 3A 0 ) - (X - d%) (if 0 + d%) = 2 {if 0 3if 4 - A 4 3A 0 },
where the arguments are u, v, and the 3 denotes at 4- + v ~.
au av
Also AnAj + if 4 if 5 , that is, if 5 (u + u) ^ {u — v!) + ^ (u + u') % (u — u'), becomes simply
= 2^^, and similarly if i 3 if 9 + if 9 if i3 becomes = 2A 13 if 9 ; and the equation thus is
- 2a 1 7 1 (if 0 3if 4 - if 4 3if 0 ) + (a^ + y/d/3 4 ) %% + (- + y/d/3i) A 13 A 9 = 0,
where the proper suffix 1 is restored to the a, 3/3, 7, and 38.
139. The equation shows that the differential combination if 0 3if 4 — if 4 3if 0 is a linear
function of A 5 Ai and if 13 if 9 , the coefficients of these products being of course linear
functions of u and v. Writing the equation
if 0 3if 4 — if 4 3if 0 = A if 5 A, + I/if 13 if 9 ,
we can if we please determine the coefficients in terms of the constants c, c", c", c lv , c v ;
viz. taking u, v indefinitely small, we have
if 0 = c 0 , 3if 4 = u (cl" u + cl y v) + v (cj v u + c/v),
if 4 = c 4 , 3if 0 = u (c 0 '"u + c 0 iv v) + v (c 0 iy u 4- c 0 v v),
Ai = Ci, if 5 = c 3 u c 3 v,
if 9 = C 9 , if 13 = Ci 3 u H- c 13 v,
or substituting, and equating the coefficients of u and v respectively, we have
c 0 (cl" u + cl v v') - Ci (ic"'u + c 0 ‘V) = Acicl + Bc 9 c 13 ',
c 0 (ci v u + c 4 V) - c 4 (c 0 'V + Cqv) = Acicl’ + Bc 9 Ci ",
which equations give the values of A, B.
140. Disregarding the values of the coefficients, and attending only to the form
of the equation
if 0 3if 4 — if 4 3if 0 = Aif-Ai + I/if i 3 if 9 ,
