66 
Proceedings of Royal Society of Edinburgh. [sess. 
on the ground that we have the identity 
M = MKM y - IVil} + «i (KM* - I Mol}. 
the term a 1 { | af l \ x — \ bf^ | } being subsequently left out of 
both denominators for the same reason as before. The result thus 
reached is consequently 
KV 
I a A i y ~ i a A i 
I Vo I - I a o b i I x 
I I - I a A I y 
or, if we write £ , r; for the values of x , y which make u 0 - t 0 = 0 , 
u i " h = °> 
a 0 b 
i 0 t 0 u 0 
1 1 
+ 
vh 
+ 
h - «i/ 
+ 
: 
1 i ^ 
oubly-infinite series here are 
tjT t" t v v 
UP+ 1 ’ US+ 1 ’ x m + 1 ’ y v +' ’ 
we deduce 
I a o b i 
/ mj. n 
°1 
t.e., 
u Q m + 1 u 1 n+1 
i^rj v 
x^ +1 y v+1 ’ 
j mj n 
L 0 
‘ eLi (< a 0 x + b Q y)™+ 1 (b 1 y + a 1 x ) n + 1 
! Vo 1 M * 1 Vi 1 v 
| a 0 b Y | < a +"+ 1 • x^ +1 y v+1 
} 
where m, n on the one side and g, v on the other are to have all 
integral values from — oo to + oo . Since the coefficients of 
t-tfjxHf on the two sides must be equal, we obtain the theorem : — 
