802 
MR, J. W. L. GLAISHER ON RICCATI’S 
32. It follows from the forms of u in art. 29 that 
1 cl V +1 
x i+l [- — ) e ax =Ax '(- 37 ) e a '\ 
1 d\~i 
x dx 
and it can be readily verified, that A = a &+1 , so that we have 
x 2i+1 (- -rr ] e! Jj: = ct 2i+1 f - —) e ax 
x dx 
1 d\-i 
x dx) 
(24). 
Transforming this result by putting x=qz? (q unrestricted), it becomes 
If now q— ^r p , this may be written 
d\— « 
a \ 2 a -z 
z\ z 2q e* z9 =aa(z 2,1 & z \ 
and, on putting q =———- we obtain the same result; so that this formula holds 
AI "i X 
good whenever q is of the forms iyry-y. 
It follows from this theorem and from the two formulae at the end of the last article 
that 
d 
z[ z~ z v +1 ~) 2,1 =m/z -2 ^ +1 ^ 2q ei' 
=&**-*{ i _w i + ( wxi=3¥)(a. y_ & , 
q Sazt q.2q \8 az?j ' 
where q=±^~ r 
The relation (24), or, as it may be written more conveniently, 
aA +1 
1 A' 
x dx 
i +1 
e ai '=- a~ i+l e ax 
(25), 
admits of being established as follows. 
Suppose e nx expanded in ascending powers of x, and consider the term in x p : we ha.ve 
