48 MR. ROBERT RAWSON ON DIFFERENTIAL 



fc+fiM+l; ..... (31) 



Ax 2n + l Aj;2»+i 



2a} 



y^ + ay + -e 2W+I -— e 2M + x =0. . . (32) 



Equation (30) coincides with the Riccatian form in one 

 case only, viz. when the exponent of x is zero. 



This property vanquishes 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 so- 

 lution of (30). 



Assume the Riccatian 



du . z 



dt +w=1 - 



Change the independent and dependent variables t and w, 

 for x and z > by the equations 



3(2ft+i)£=AV +I ; 



3Pz=A*x zn w-^. 

 or 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«V' + 3RV + 3S=o . . . (33) 



be a general cubic where (a) is constant, and R, S func- 

 tions of x } y. 



Differentiate (33) with respect to x } y respectively, then 



(V + *V + R)g + «:v+g-o,. • (34) 

 tV+^V+B)^* V+5-* . . (35) 



