EXPLICIT FORM OF THE COMPLETE CUBIC DIFFEBENTIAL RESOLVENT. 439 



to have breathed into it a life of such vigour that it will flourish for 

 ever. 



On the Explicit Form of the Complete Cubic Differential Resolvent. 

 By the Kev. Egbert Harley, F.R.S. 



[A communication ordered by the General Committee to be printed m extenso 



among the Keports.] 



This paper is intended as supplementary to others relating to the 

 theory of differential resolvents which I have had the honour to sub- 

 mit to the Section at former meetings of the Association (see ' Reports,' 

 Transactions of Sections, 1862, pp. 4, 5 ; 1865, p. 6 ; 1866, pp. 2, 3 ; 

 1873, pp. 17-21 ; 1878, pp. 466-8). 



About four years ago Mr. Robert Rawson and myself calculated by 

 independent methods the complete cubic differential resolvent ; in other 

 words, we determined the explicit form of the linear differential equation 

 of the second order which is satisfied by any root of the general algebraical 

 equation (with unmodified coefficients) of the third degree. The result 

 at which, after much labour, we both arrived has not hitherto been 

 published ; and I desire now to place it on record, indicating at the same 

 time some of the details of my own calculation. The process employed 

 by Mr. Rawson may be elsewhere explained. 



Write the cubic equation in the form 



and consider the coefficients a, b, c, d as functions of a single parameter, 

 say X. Differentiate with respect to x, and denote the differentiations by 

 accents ; then a slight reduction gives 



^,^ (a'h—ah') i/ + (a'c-ac') y+^(a'd—ad') . 

 ^"^ a(a7/ + 2by + c) 



Integral ise the function of ?/ by any of the known methods, the result 

 takes the form 



Ay'= {(a'b - ab')y^ + (a'c—ac')y 

 +^(a'd-ad')} (By^ + Cy + B), 

 where 



A=a(aW-6abcd+4-ac^ + UH—3b^c^); 

 B=2a(ac-b^) ; 

 C=-a''d+7abc—6h^; 

 D=-abd + 4<ac^-3b^c. 



Develop the right-hand member and eliminate aU powers of y higher 

 than the second by means of the original cubic. 

 We thus find 



Dy'='Ey'' + Fy + G (1) 



in which 



D =aW-6abcd-i-4>ac^ + 4<¥d--3bh^, 



the cubic discriminant; 

 E = 



