DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 455 



104. In the case when the invariant 93( = Q 3 ) vanishes, so that the quartic is 

 canonically binomial, the equation in v is linear and of the fifth order only,* being 



dv 



and there is, therefore, a linear homogeneous relation among the six quantities v. 

 The constants in this relation depend partly on the choice of the fundamental 

 system of integrals, partly on the invariant Q t ; e.g., for the equation 



d*u , c 



^ ? 

 we may take 



M 1 = Z"', U 2 = Z-, U, = Z-, W 4 =Z"S 



where 



2m, = 3 - {5 - 4 (1 - c)}, 2^ = 3 - {5 + 4 (1 - c)}, 



2ro 2 =3+{5-4(l-c)}', 3m 4 =3 + {5 + 4(1 - c)}*, 

 the indices m lt m 2 , m^ m 4 all being roots of 



m (m 1 ) (m 2) (m 3) + c = ; 

 and the linear v relation is then 



Multiplying the equation by v, it can be integrated once, with the result 



d*v dv cPv 



where A is a determinate constant. This constant depends, like those before, partly 

 on the choice of fundamental integrals and partly on the invariant 4 ; and it changes 

 from one quantity v to another. Recurring to the particular example, we have 



v u = {5 4 (1 c)}2 8 = 0z* t 

 say ; and, substituting, we find 



20* = 2c0* + A 

 HALPHKN, ' Acta Math.,' vol. 8, p. 329. 



