DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 463 

 Hence we have the result that, if 



[or, *] = 0, [a, *]. = 0, [z, x]f = 0, 

 then 



[<r, x\ = 0, 

 where 



113. But, on the other hand, while the algebraical relation between <r and x 

 involves the proper number of arbitrary constants, they are not in general equi- 

 valent to 2p I independent constants ; for, by the method of construction of cr, 

 all the constants which enter into its expression are composed of the other 

 (2m 1) + (2 1) -f (2p 1) independent arbitrary constants, a number in 

 general less than 2p 1. There is therefore not, in general, a justification for an 

 extension of the result so as to include its converses in the form that, if any three of 

 the derivative equations be satisfied, the fourth is satisfied ; and it is only when there 

 are certain coefficient-limitations on the form of <r as an algebraical function of x that 

 the converse can be asserted. An illustration will be given in 118. 



114. The simplest case which occurs is that in which m = 2 and p = 2; for 

 then p = p, and the deduction to be made is that, if 



[, *1 = 0, 

 then 



+ d' gz + 



where a, b, . . ., h are constants. The converse is also tme ; for in homographic 

 transformations an interchange of the transformed variables leaves the functional 

 character of the transformation unaltered. Since then these homographic transfor- 

 mations do not alter the order of the derivative equation, we are led to investigate 

 the modification caused by them in the derivative itself. 



Considering then, the ntic derivative [s, z] g , let us find the effect of a homographic 

 change of the independent variable given by 



* 



*= ^ = a;(z) 

 in CAYLEY'S notation. Now, as in 11, we have 



dz" " rml r\ daf 

 where 



