of Edinburgh, Session 1875 - 76 . 
97 
n i = n i • 
Again, just as the solution of this equation has the property 
c p _ c p + q 
€ c — e , 
so it is easy to see that we have in (8) 
Ap) /(?) = ZO+2)i 
where the bracket over p + q is employed to indicate that in the 
expansion we must square the numerical co-efficients of each term 
of a power of this binomial, i.e ., 
P + q = p + q , 
p + q 2 - p 2 + q 2 -\- 2 2 pq , 
p -f- <f = p 3 + f -f- 3 2 ( p 2 q -f pq 2 ) , 
p + f = p 4 + + 4 2 ( qpq + p$ 3 ) + 6 2 p 2 ^ 2 , 
&c., &c., 
and a similar property, though of course involving higher powers 
of the co-efficients, holds for each of the functions IL m above. 
For the product of any two similar expansions (with different 
variables) is easily seen to have all its numerical co-efficients raised to 
any given power when those of the separate expansions are so raised. 
The paper contains also an account of various attempts to solve 
the general equation of the second degree, of which the following 
may be noted. 
a. Transform to 
S- x ^=°’ 
and evaluate 
JJ y tlx 1 
at once , just as 
r^dx 
J y ax 
is evaluated. 
The difficulty is reduced to finding the value of 
where a sinyle operation is to be effected. 
VOL. IX. 
0 
