317] 
ON THE TRANSFORMATION OF A CERTAIN DIFFERENTIAL EQUATION. 
79 
whence the transformed equation in 6 must be this very equation, that is, it must 
be the first equation. I have for shortness used the particular integral (Vl + x 2 + xf m ; 
but the reasoning should have been applied, and it is in fact applicable, without alter 
ation, to the general integral 
C (Vl + x 2 + x) m + 0' (Vl + o? — x) m . 
There is of course no difficulty in a direct verification. Thus, starting from the 
first equation, or equation in 6, the relation id = 2x- +1 gives 
dy _ i dy d 2 y _ i d f i dy\ _ 1 ¡dry 1 dy\ 
dd 4x dx ’ dd~ 4x dx \4x dx) lQcc 2 \dx' 2 x dx) ’ 
1 + 6' 2 = — 4x 2 (1 + x' 2 ) ; 
so that the equation becomes 
1/1+** d A + M dy_ 
4 War* x dx) 4x dx ‘ 1 
or multiplying by 4, 
that is 
d-y / 1 + x 1 1 + 2x-\ dy 
(1+it2 >s?+ 
+ 
dx 
4 m 2 y = 0 ; 
< - 1 2 += °’ 
the second equation. But the first method shows the reason why the two forms are 
thus connected together. 
2, Stone Buildings, W.C., February 19, 1862.