DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 449 



r -I/*)' 



or 



=r = (^'-*P,)v3 



and 



and the dependent variables are connected by the relation 



y = ttv'. 



93. There are many special cases of these forms depending on simpler analysis ; 

 thus, one of such cases is that wherein the priminvariant 6 3 vanishes, and then the 

 form (27) comes to be binomial, while in (28) the coefficient R^ is constant. 



The two forms (27) and (28) are practically the alternative normal forms of the 

 quartic ; it is not possible by this method to reduce the general equation to the 

 binomial form 



for such a reduction requires that the coefficients of cPu/dz?, d?u/dz z , du/dz shall all 

 vanish three conditions which cannot, in general, be satisfied by proper determina- 

 tion of the multiplier X and the independent variable z. In the case of all the forms 

 which have been chosen the general assumption has been made that it is desirable to 

 remove from the equation the term of order next to the highest ; for any equation, in 

 which this might not be done, other forms could be obtained, but the analysis of 

 Section II. shows that those forms adopted have the advantage of being most easily 

 obtained. It may be remarked that, for the reduction of any equation to the 

 canonical form adopted, the subsidiary equations are all of order less than that of the 

 equation to be transformed. 



The Quotient- Equation for the Quartic. 



94. The differential equation satisfied by the quotient of two solutions of the 

 quartic must be of order 7 (= 2.4 1), since each of the solutions contains implicitly 

 four constants in linear and homogeneous form. 



Taking u x and 3 as two particular solutions of the equation in its canonical form, 

 and denoting their quotient by //,, we have 



"2 = %/* 5 



proceeding as in the corresponding case for the cubic, the following equation is 

 obtained, viz. : 



MDOCCLXXXVIII. A. 3 M 



