ON DIFFERENTIAL RESOLVENTS ETC. 41 



VI. On Differential Resolvents and Partial Differential 

 Resolvents. By Robert Rawson, Esq., Assoc. I.N. A., 

 Hon. Member of the Society. 



Bead January io, 1882. 



Arts, i to 8 contain the first and second differential re- 

 solvents of a cubic, using therein one independent variable 

 only ; and from the second differential resolvent the usual 

 root of the cubic is determined. 



In arts. 9 to 11 there is the first partial differential 

 resolvent of a general cubic, using therein two independent 

 variables ; and from this partial differential resolvent the 

 root of the general cubic is obtained. 



Arts. 12 to 14 contain the first partial differential re- 

 solvent of a quartic, using therein three independent vari- 

 ables ; and from this resolvent the root of the quartic is 

 determined. 



1. Let 



y^ + ay + m—o, (1) 



be a cubic in which (a) and (m) are functions of x, a vari- 

 able parameter. 



For convenience, z/ T =-^, a T = — , &c, will be used. 



Differentiate (1) with respect to oc, then 



y'=- a -^^=vf+Qy+K, ... (2) 



