NOTE ON THE TRANSFORMATION OF A CERTAIN DIFFERENTIAL 
EQUATION. 
[From the Philosophical Magazine, vol. xxiii. (1862), pp. 266, 267.] 
The differential equation 
if we put therein id = 2a? 2 + 1 (i = V — 1 as usual), becomes 
(1+a ' s) S +ic tC 4M ^ =0 ' 
In fact an integral of the second equation is (VT + a? 2 -fa?) 2 ” 1 ; this is 
= (V(2a? 2 + l) 3 — 1 + 2a? 2 + l) w ; 
or putting id = 2a? 2 + 1, it is 
= (V 1 +id) m , 
which is 
= {» (Vd 2 +1 + 6)\ m ; 
so that an integral of the transformed equation in 0 is 
= (Vd 2 + 1 + d ) m ; 
and writing in the second equation 0 for a?, and \m for m, we see that the last- 
mentioned function, viz. (*Jd- +1 + 0) m , is an integral of 
O + P)% + ef e -r>i'y=0;