12 MR. ROBERT RAWSON ON DIFFERENTIAL 



where P, Q, R are functions of (a, m) to be determined as 

 follows : — 



— a'y— m 1 = 3P2/ + 3Q?/ 3 + (3R -f aV)y z + aQy + aR. 



Eliminate from this equation y* and y z by means of the 

 cubic (1), then 



— a x y— m , = 3P(— ay* — my) +^Qi{—ay—m) 



+ (3'R + aV)y i + aQy-\-an } 

 or 



(3ft — 2aV)y z + (a 1 — 3mP — 2aQ)y + m x + aR — 3mQ=o. 



To satisfy this equation it is necessary and sufficient that 



3R-2aP = o, (3) 



a 1 — 3mP — 2aQ=o, (4) 



m l + dR — 3mQ=o (5) 



From these equations the values of P, Q, R are readily 

 found to be 



P(4« 3 + 27m 2 ) = gma 1 — 6am l , 

 Q (4a 3 -I- 27m 2 ) = 2a V + gmm\ 

 R(4c 3 + 27m 2 ) — bamd 1 — \a?m* ; 

 .-. 3R = 2«P. 



Substitute these values in (2), then 



. qma x — 6am* , 2a 2 a l +qmm x 6ama* '— A.a z m T ,,. 



v = — .v H . ?/ H z (6) 



v 4.a z + 2jm z * 4a* + 27m 2 J 4^ + 27m 2 v ' 



Equation (6) is therefore the first differential resolvent of 

 the cubic (1). 



Hence the cubic (1) is the integral of (6), and the value 

 of?/ which satisfies (6) is then a root of the cubic (1). 



