810 
MR. J. W. L. GLAISHER ON RICOATI’S 
37. As mentioned in art. 34, the integral 
1 / q d V/Cje ax + Cc,e ax 
U ~ri+i\ X dx) \ 
of the differential equation (1) was first given by Boole in the £ Philosophical Trans¬ 
actions ’ for 1844. 
The integral 
1 d\ l fc^e™ + Cc£- ax 
x dx) \ x 
is due to Mr. Gaskin, and was in effect given by him in a problem set in the Cam¬ 
bridge Senate House Examination for 1839. The problem is as follows* : 
“ If m be the greatest root of the equation m 2 +m=a, 
cd m or Cx » + i 
r - n [ x m \/r 
(f —[ Vr 2 — n 2 ) m cos (nc-J-a) 
\J r=n J r=—n) 
are general values of y in the equation d}y-\- i^n 2 — 0 jy= 0 according as m is an integer 
or fraction : and in the first case (dJ lJ r n 9 ') m+1 u= 0 where u — yx m ; apply the first or 
third result to solve the equation 
dj l y+(n*--)y=0.” 
Thus Mr. Gaskin’s theorem is that the solution of 
is 
dhi, 0 p(p + 1) 
ax x* 
u 
U— Cx P 
' d \p cos (x\/r + «) 
dr) \A 
(37), 
where r is to be put equal to o? after the performance of the differentiations, p being 
a positive integer, and that in general 
i; = Cx ?,+1 | (r 3 — oF) p cos (rx-\-ct)dr .(38), 
p being any positive quantity. 
* The problem forms the second part of Question 8 of the paper set on the afternoon of Tuesday, 
January 8, 1839 (‘ CambridgeMJniversity Calendar,’ 1839, p. 319). 
