36 
The solution of the equation 
($)-w 
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 
22" — 2 = x 
= 
Differentiating this again, we find 
zz” -2z2' =0, 
the integral of which is 
2’+a’z=0; 
therefore, 
2e(P' 4 a} 0, 
and 
= {(D + at) 0}F 
5: To exemplify this theory we may assume yf = 2”; 
yp a m Wigs yee 5 + _¢ 
whenee ( gee, teeth m, and @— ¢ =¢, 
The general formula becomes, therefore, in this case 
{ (D+ 9) (D+ p+a-5)- Caese ee 
By making c = 0, and writing - m in place of 2, this be- 
comes 
{(D+9)(D+ 9+ o) ~ MRD) 92x, 
which is equivalent to a general soluble form which Dr. Har- 
greave has obtained by an entirely different method.* 
* Phil. Trans., 1848, p. 35. 
