DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 469 

 By the transformations of 1 1 the foregoing differential equation becomes 



h . . . = v, 



mm 



where 



and, in order to make the term in d*~^ujd^~ } disappear, the relation Xz' 4u ~ u = 1 has 

 been adopted ; hence we have 



The variable z is at our disposal ; and, though in the general theory a choice of z 

 fundamentally more effective than the following can be made (as was done in 29, 30), 

 yet, our present aim being the deduction of the quotient-derivatives, we shall here 

 assume that z is so chosen as to make V a constant a, a choice which appears to render 

 most simple the required deduction. We then have 



and the equation takes the form 



Let ft be the quotient of two solutions, say t and w 8 , of this differential equation, 

 so that 



t/ 1 = P + A 1 U ) +A a U 8 + ... 4 A.U., 



u, = P + B x U t + B, U 2 + . . . 4- B. U., 

 w, = j 11. 



Then the differential equation satisfied by p. is of order 2 ; and it can be obtained in 

 a manner similar to that employed in 85, 94. 



The quotient-derivatives will be obtained for correspondingly limited forms of 

 differential equations, viz., those in which the left-hand side is constituted by a single 

 term, which is that of highest order in the differential coefficient. 



120. Example I. For the equation of the first order 



we have, since "_. " ,//, the equation 



a = IJM 4 



or 



= (/x I)a4- 



