44 MR. ROBERT RAWSON ON DIFFERENTIAL 



Then (10) may be written 



Restore the values of p and Q, then 



<Py 



dx 



C d , 7 t md l — 2am l \dy C tfm^—m^a* 1 

 \dx g V4« 3 + 27m a J dx \2arn 1 — $ma l 



+ 6am(a i y-6a(m j y-6a z m l {a l ) z \ ,. 



(2am 1 — 2ma I )(4.a z + 2jm 2 ) j 



Equation (12) is therefore the second differential resol- 

 vent of the cubic (1). The value of y which satisfies (12) 

 is a root of the cubic (1) ; and either of the three roots of 

 the cubic (1) is a particular solution of the differential 

 equation (12). 



3. When (a) is constant, then the first and second dif- 

 ferential resolvents respectively become 



6am 1 dx * 2a v 3 v OJ 



d*y__(d, / — 2am 1 \\dy 3(^ , ) a _ , v 

 d^~~\dx ° g \ \/^ + 2ym i )jd^ \a> + 2ym z ' y ~ X ^ 4 ' 



4. If (m) be such as to satisfy 



m 1 , . 



«*—r h (IS) 



then the coefficient of -r in (14) is zero, and it becomes 



dx 



^-2-o (l 6) 



dx 2 - 9 • (IDj 



