818 
MR. J. W. L. GLAISHER ON RICCATI’S 
Thus from (52) and (53) 
yj v=4 i .z! j x~ l e~ w cli 
’o 
= 4 ‘.ill —~ v, from (51), 
2 a da 
which is the relation (50). 
It follows from (50) in connexion with (21) of art. 31 that 
a formula given by Hargreave in the ‘Philosophical Transactions’"" for 1848, p. 34. 
42. In the paper just referred to Hargreave obtained by a symbolic process the 
solution of the differential equation in the form 
and thence, by (54), deduced the solution in the expanded form. 
Hargreave also gives on p. 45 of his memoir the complete solution of the equation 
d~u 2m . 
7 -+——era = 0 
dx- x 
in the form 
u-—c x j (z 2 — dr) m l e ,: ~dz -fi eye 3 “ +1 ( 2 3 — cr) m e x: dz 
= Cj[ (z 3 — l) m ~ 1 e axz dz-\-CciX~ %n+l [ {z?—l)~ m e ax: dz. 
J _ i J _ i 
J -1 
. . (55). 
One or other of the definite integrals in (55) is however always infinite, except when 
to lies between 0 and 1. 
In the case of the differential equation (1), this solution becomes 
u=c 1 x~ J, { (z 3 — l)"^ _1 e ai -+c^ ;+1 ( (z~—iye' u: dz, 
" J-i 
or, as it may be written more conveniently, 
u=c 1 x~i (1— 2 3 ) -i,_1 e^+c s a^ J+1 | (1 — z^)Pe ax “dz. 
* “ On the Solution of Linear Differential Equations,” pp. 31-54. 
