846 Mr. Jerrard on the Occurrence of the Form g, 



minate when m is general in value. If then it should appear 

 that for a certain value of m the equation 



f/ = F + P''a: + P'"^ ... + ?("») x'"-'^ 

 which arises from combining the equations (1.) (2.), would 

 take the form 



= + Oo: + 0a?2,.. _j- Ox*^-\ 



in what light ought we to regard this result ? The expression 

 for X will most certainly in this case assume the form 



but ought this to be regarded as the ultimate form for x ? 



May not the expression —- become interpretable by following 



the known methods for the treatment of such functions ; or 

 rather, must it not be susceptible of an ulterior form ? 

 To fix our ideas, let 



7» = 4, A = 4, 

 also let 

 P=/aS), R=/,(S), Q=/3(S), y=f,{S), T=l. 



Accordingly we shall have 



^ _ *jsi 



Then if 



7^ = 0, F = 0, P" = 0, F" = 0, P'^ = 0, 

 or, 



/4(S) = 0,/i(S)-D = 0,/3(S)-C = 0,/2(S)=0, S=0; 

 that is to say, if the functions designated by fx^f<i^f^,f^ be 

 such that 



/4 (0) = 0, / (0)-D = 0, /3 (0)-C = 0, /s (0) = 0: 

 ought we to conclude generally that 



will not admit of a definite interpretation when 

 S = G, 4» (0) = 0, * (0) = ? 



If all the quantities y, F,P", ... P^*"^ could be shown to be 

 determinate in value, and any one of them except y and P' 

 not equal to zero, it would not be necessary to proceed to the 

 ultimate form of x^ in order to show the possibility of ex- 

 pressing X by a determinate function of 3/, P, Q, R, ... L 

 involving the solution of an equation of less than m dimen- 

 sions. 



