208 
Mr. Babbage’s essay towards 
if we suppose a ( 1 , 1 ) = (3 ( 1 , 1 ) these equations become 
^ 2 {x, y) — (p 
v 8 (£> A . 
‘<?’A+A 
(5- A / 
d- 3 ’ 3 (-z,.)’)=<P 3 ^ 
and generally we should find 
= ( 6 ) 
where ^ (a:, y) — <P ancl also « (i. i) = 0 (l , 1 ) 
A more general expression, and one which contains (b) as a 
particular case may be deduced in the following manner. 
Let ^(^y) = {^<pg|l}” = m K 
taking the second simultaneous functions on both sides we find 
. — (r V v 5 fi (i, i) a « o> qeu ” zr \ K n ~ m \ n ~ 
V (X’y)— {7(777) ^ /s (I, I) K* I — l.( I, I) ^ |S ( 1 , 0 i — 
—5 Altilmf »P-Qg('>0 o * (x -- n \ iTr f ^2 J *£-il ^ 1 " -m 
and if we continue to take the succeeding simultaneous func- 
tions we shall find generally, that when 
i 
» 
**»<«> -{*x^-)r <»>. 
this expression is much more general than the preceding ( b ) 
