18 REPORT 1873. 



losophical Magazine,' the 'Quarterly Journal of Mathematics,' the _' Manchester 

 Memoirs,' and the ' Proceedings of the London Mathematical Society.' It has 

 been shown that every dilierential resolvent is satisfied, not only by each of the 

 roots, but also by each of the constituents of the roots of the algebraic equation to 

 which it belongs, and that these constituents are in fact the particular integrals of 

 the resolvent equation. In the latter aspect every differential resolvent of the form 



t+cp(D)e'"'u=0 = V,\j) = ^^, 



in which is a variable parameter, and u considered as a function of 6 is a root 

 of a certain algebraic equation of the (« + l)th degree, gives, when U is of an 

 order higher than the second, a new primary form — that is to say, a form not re- 

 cognized as primary in the late Professor IBoole's theory. And in certain cases in 

 which the dexter of the defiuina: equation does not vanish, a comparatively easy 

 transformation will rid the equation of the dexter term ; and the resulting dif- 

 ferential equation will be of a new primary form. The same transformation which 

 deprives the algebraic equation of its second term will deprive the dift'erential 

 equation of its dexter term. 



Boole, in his last paper before the Roj'al Society, entitled " On the Differential 

 Equations which determine the form of the Roots of Algebraic Equations,'' re- 

 marks : — " While the subject seems to be more important vnih relation to differ- 

 ential than with reference to algebraic equations, the connexion into.which the 

 two subjects are brought must itself be considered as a very interesting fact. 

 As respects the former of these subjects, it may be observed that it is a matter 

 of quite fundamental importance to ascertain for what forms of the function 

 ^(D), equations of the type 



admit of finite solution. We possess theorems which enable us to deduce from 

 each known integrable form an infinite number of others. Yet there is every 

 reason to think that the number of really primary forms (of forms the knowledge 

 of which, in combination with such known theorems, would enable us to solve all 

 equations of the above type that are finitely solvable) is extremely small. It will 

 indeed be a most remarkable conclusion, should it ultimately prove that the forms 

 in question stand in absolute and exclusive connexion with the class of algebraic 

 equations here considered." (Phil. Trans, for 1864, p. 733 ct seq.) 



In his later researches the author of this paper has sought to determine the 

 form of the differential resolvents of algebraic equations whose terms are complete, 

 and whose coefficients are unmodified. Mr. Spottiswoode has also considered the 

 question in this its most general aspect; and in a short paper on "Differential 

 Resolvents," printed in the second volume of the third series of the ' Manchester 

 Memoirs,' pp. 227-232, he has exemplified a method of finding the resolvents in 

 the cases of quadratics and cubics, which is directly applicable to all degrees. 

 This method, considered as a working process, possesses some advantages over that 

 employed by Sir James Cockle and Mr. Harley in dealing with trinomial forms. 

 Its chief peculiarity consists in effecting all necessary eliminations by means of 

 determinants. 



Beginning with the quadratic 



(fl, b, c) {X, iy-=o, 



which gives 



2(a, b) (.r, 1) .r' + («', b', c') (x, 1)2 = 0, 



where differentiation with respect to the parameter is indicated by accents, Mr. 

 Spottiswoode forms a system of equations from which by the elimination of all 

 powers of .v higher than the first, he deduces 



