[419 
420] 
9 
420. 
LTORS. 
ON BICCATFS EQUATION. 
which occur in the 
1866), pp. 461—472, 
[From the Philosophical Magazine, vol. xxxvi. (1868), pp. 848—351.] 
as follows: 
The following 1 is, it appears to me, the proper form in which to present the 
solution of Riccati’s equation. 
The equation may be written 
+ V 2 = x lq ~ 2 , 
ax d 
which is integrable by algebraic and exponential functions if (2i + 1)^ = ± 1, i being zero., 
or a positive integer. To effect the integration, writing y = - ~ , we have 
'll/ QjX 
ax 1 
The peculiar advantage of this well-known transformation has not (so far as I am aware) 
been explicitly stated ; it puts in evidence the form under which the sought-for function 
y contains the constant of integration. In fact if u = P, u = Q be two particular solutions 
of the equation in u, then the general solution is u = CP + DQ; and denoting by 
P', Q' the derived functions, the value of y is 
B x , 8 y , and P for B 
ly operator such as 
?hus E^E^*, if we 
*, i.e. (E 1 *)x(E 2 *), 
OF + DQ' 
y ~ CP +DQ ’ 
showing the form under which the constant of integration C -4- D is contained in y. 
To complete the solution, assume 
1.F« 
u = ze q ; 
C. Vil. 
2