and on Go-resolvents. 185 



whence, by equation (9), 



u2=0^= GC.w'' - - - -(25) 



and r . u = M %v (26) 



M being a constant. The analogy between the algebraical 

 and differential theories holds then thus far : the supposition 

 that u or, which is the same thing, ^u is known, is the same 

 as supposing that the solution of (1) the given equation, is 

 known, and consequently to effect the transformation we 

 have to encounter all the difficulties of solving the original 

 equation. The analogy fails thus far : if we can solve (1) 

 we can annihilate its final term. This failure of analogy 

 seems analogous to another failure of analogy between 

 quadratics and linear differential equations of the second 

 order : a quadratic may have equal roots, but there are two 

 arbitrary constants in the complete integTal of every linear 

 differential equation of the second order. 



§ 2. On Co-resolvents. 



21. The Theory of Co-resolvents originated in my "Sketch 

 of a Theory of Transcendental Roots," published in the Fhilo- 

 sophical Magazine for August, 1860. The subject of that 

 paper has since been pursued as well by myself as by Mr. 

 Harley, Professor Cay ley, Mr. Russell, Mr. Rawson, Mr. 

 Spottiswoode, and the lamented Boole. When two or more 

 conditions involving a quantity are simultaneously satisfied 

 by the same value of the quantity, those conditions, and 

 indeed that by which the value is determined, may be 

 termed " Co-resolvents," and one co-resolvent may sometimes 

 not inaptly be termed a ''resolvent" of another. The 

 co-resolvents may be 'algebraical resolvents,' or 'differential 



resolvents,' or 'functional resolvents.' Thus the 



three 



equations 







y^—^y^'2x = 



- 



- (i) 



S^(l-.^)J|-3^4^,^ 



= 



- (ii) 



dct>{x) ^{^fl-x') ^ 

 dx ^sfl—x' 



- 



- (iii) 



- i 



