240 
Proceedings of the Royal Society 
Applying tlie same principle to (2), we shall also get 
hr, . h„_^ + Zp\ . . hr,.^ + f/ . 
= (p+?)^[c:; 
and in like manner 
while generally 
K -=p, . K.r + 
+ . . . + - 2r 
= (p + <2)’’ . [7^]'*]^’^ symbolically . . . (3). 
Again, hr, may he expressed under another form — in terms of the. 
coefficients of the equation (p and q) only. Here it will be; 
necessary to distinguish the cases n even and odd. 
We have 
lh^plh + q=p^ + q, • 
\ =p!h + • 
By repeated applications of formula (1) we may deduce 
Ag = + 7 p^q +15 phf + 10 p^^q^ + , 
/ig + ^P^q + 21p'^2^ + 20p^2^ + bpq^ , 
and generally 
K, =F’' + (‘in-\).tr’'-\q + (2» ~ 2) (2w - 3) 
(2«-3)(2«- 
4)(2». 
l)n 
^ 1,2 
. 3 
. p 
?+...+ ^ 
2 ' 
■ V 9. 
+ q’‘ . . . 
0), 
and 
+ 
1 
1)(2m - 2) 
1.2 
. . q^ 
{2n 
- 2) (-2« - 3) (‘hi 
~ 4) 2m-5 ^3 I 
(n + '2)(n + 
l)n 
"V 
1.2.3 
• t' 
' '1 • • • < 
1.2.3 
p q 
+ (n + \)pcf . . (5), 
each of which series consists of ?^ + 1 terms. 
