DDIUVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 447 



quotient of two solutions of this equation of the second order, and taking one of these 

 solutions to be j, we have, by the application of a well-known formula, 



dz "itjV 

 But, by the result quoted in 80, we have 



_iii J 



t ^ v i ** 



so that a combination of the results obtained gives the solution of the equation in p, 

 which is 



Equation of the Fourth Order. 

 91. The general results obtained in 30 show that, by the solution of 



{*,*}=*P 



the equation 



% + ?.% + r.2 + *.-o 



is transformed to 



+4Q,t+Q.u = ....... (27), 



where 



z /l y = u, 

 and 



. .. 



2 dz 



and, if we write 2' = 0~* t the equation determining 6 is 



The form (27) is a canonical form of the quartic in conformity with the general 

 canonical form ; and the quartic can be reduced to this form by the solution of a 

 linear equation of the second order. 



