36 
The solution of the equation 
(a 
may be obtained by putting y = 2* where s is as yet indeter- 
‘minate; thus it is reduced to 
Ss () = hz", 
z 
And if we now determine s by making rs = 2, we get 
x rk 
dead fr a 
2 
Differentiating this again, we find 
ze’ —- 22 = 0, 
the integral of which is 
2 +a°z=0; 
therefore, 
z=(D +a’) 0, 
and 2 
y= (D's ay 0) 
5. To exemplify this theory we may assume yf = 2”; 
Cha Wei oh 7 c 
whence (7 see r=, h=—m,and@-$ =e ——. 
The general formula becomes, therefore, in this case 
c\ (n+1)(e+n) N 
By making c = 0, and writing - m in place of n, this be- 
comes 
{(D+9) (D+ 940) - 21 yex, 
which is equivalent to a general soluble form which Dr. Har- 
greave has obtained by an entirely different method.* 
* Phil. Trans., 1848, p. 35. 
