64 
An equation which is soluble by means of (26) and (28) for 
all values of the function r. 
8. Put, r = Ax 2n , where (A) is constant, then 
dz n(n + 1) A 2 . 
Tx + ^ z ~ /to 2 + 9/3* 
which is soluble by means of (81) and (82) 
dy _ n 
-~ + 6z + - 
ydx ' x 
y* + ay + Tje 
A# 2n+1 Ax 2n+1 
2^+1 2a 3 2 n + 1 
27 € 
= 0 
.(30) 
.(31) 
,(32) 
Equation (30) coincides with the Riccatian form in one 
case only, viz., when the exponent of x is zero. 
This property vanquished all hopes of connecting the 
solution of Riccati’s equation with the roots of a cubic in its 
present form. This result was communicated to Sir James 
Cockle, who kindly sent me the following neat solution 
of (80). 
Assume the Riccatian 
Change the independent and dependent variables t and u, 
for x and z, by the equations 
3(2 n -i-l )t = A h x 2n+1 
3Bz = A*x 2n w 
' x 
Then the above Riccatian on reduction coincides with (30). 
The solution of a general cubic by a partial differential 
resolvent with two independent variables. 
9. Let V 3 + 3 aV 2 + 3RY + 3S = 0 (33) 
be a general cubic where (a) is constant and R, S functions 
of x , y. 
Differentiate (33) with respect to x, y respectively, then 
(V 2 + 2aV + B)Y + ^ 
dx dx 
(V 2 + 2aY + R)Y + ^ 
dy dy 
v + f-o 
dx 
(34) 
v + f=o 
dy 
(35) 
