Annual Report of the Council. 221 



led to consider the problem in connection with some 

 researches on the finite solution of algebraic equations, in 

 the course of which he calculated the sextic by a direct 

 process. He published his researches with the details of 

 his calculation in the Society's Memoirs ; and followed up 

 the subject in two papers on the theory of quintics in the 

 Quarterly Journal of Mathematics, and also in an elaborate 

 exposition of Cockle's " method of symmetric products " 

 in the Philosophical Transactions. The study of these 

 papers led the illustrious Cayley to investigate the subject, 

 and his results were embodied in a memoir entitled 

 " On a New Auxiliary Equation in the Theory of 

 Equations of the Fifth Order," which appeared in the 

 Philosophical Transactions for 1861. Cockle had calculated 

 the auxiliary equation for one of the trinomial forms to 

 which the quintic may be reduced without any loss of 

 generality ; hence the simplicity of his result. Cayley, 

 employing an invariantive process, calculated the same 

 equation for the complete quintic, that is, the quintic not 

 deprived of any of its terms and not modified in any of its 

 co-efficients. The result is, of course, less simple than that 

 for the trinomial form, but it has the advantage of being 

 absolutely complete. Thus, Cockle's labours on the quintic 

 invested the theory with a new interest, and the methods 

 he devised, and the results he obtained largely directed the 

 course of subsequent speculation on the subject. 



His mode of dealing with the theory of differential 

 equations was equally marked by originality and indepen- 

 dence of mind. Not confining himself to the beaten track, 

 he pushed his way into unexplored regions, and succeeded 

 in bringing to light important relations and analogies 

 between algebraic and differential equations. He found, 

 for instance, that from any rational and entire algebraic 

 equation of the degree n, whereof the coefficients are 

 functions of a single parameter, we can derive a linear 



