MR. W. SPOTTISWOODE ON DIFFERENTIAL RESOLVENTS. 227 



and construct the non-modular group of ii . lo . 6 which, 

 has 120 substitutions of the eleventh order, 264 of the fifth, 

 no of the sixth, and 55 of the second order, by a method 

 similar to that above given for the group of 7 . 6 . 4, there is 

 no necessity for here amplifying what I have there written, 

 after what has been said on the group of 7 . 6 . 4. I hope 

 to return before long to this subject. 



XV. — Note on Differential Resolvents. 



By William Spottiswoode, M.A., F.R.S. 



Communicated by the Rev. Robert Harley, E.R.A.S. 



Bead Kovember 4, 1862. 

 Hr 



The following seems the readiest method of finding the 

 difierential resolvent of a given algebraic equation, the co- 

 efficients of which are functions of a single parameter. 

 Although exemplified here only in the cases of quadratics 

 and cubics, it is directly applicable to all degrees. 

 Beginning with the quadratic 



(«, b, cjx, 1)^ = 0, (i) 



and indicating differentiation with respect to the parameter 

 by accents, we have 



7.{a,b'^x,i)x' + {a\b\cyx,iy = 0y . . . (2) 



from which we may form the following system : — 



— (a, b^jv, I ) ( — 2a?') + ax^ + 2b' x -f- c = o, 



— (a, byjc, 1)0 + ax^ + 2bx + c = o, 



— (a, i^o:, 1)1 . -\- ax +b =0j 

 ^(a^byjc, i)a? +ax^ + bx . =0, 



> ■ ■ (3) 

 q2 



