220 Annual Report of the Council. 



mathematics. His papers may be grouped for the most 

 part under two heads, viz., Common Algebra and the 

 Theory of Differential Equations. In Algebra he worked 

 mainly among the higher equations ; and for many years 

 his labours in this department were inspired and directed 

 by the hope of " solving the quintic," or to be more exact, 

 expressing a root of the general equation of the fifth degree 

 by a finite combination of radicals and rational functions. 

 The problem had long engaged the attention of mathema- 

 ticians, and was attacked by the most celebrated analysts 

 of the last century with great skill and vigour, but without 

 success. In the early part of the present century, Abel, 

 the young and gifted Norwegian mathematician, attempted 

 to show that a finite solution of the problem was impossible. 

 To prove a negative, however, is proverbially difficult ; and 

 despite Abel's "demonstration," and the non-success of 

 preceding investigators, Cockle clung to the conviction 

 that what had been done for the lower equations might be 

 done also for equations of the fifth degree. He laboured 

 long and hard at the problem ; and although he failed, 

 like others before him, to effect a general solution, 

 yet his labour was not lost. He found not what 

 he sought for, but other things which amply repaid 

 the toil of effort, and he opened up new methods of 

 working and new lines of research which are of acknow- 

 ledged value in themselves. His first communication to 

 this Society contained a result which attracted much 

 attention on account of its remarkable simplicity. By an 

 indirect but ingenious process he succeeded in determining 

 the explicit form of a certain sextic equation on the 

 solution of which that of the general quintic may be shown 

 to depend. The accuracy of this sextic or " auxiliary " 

 equation (whose co-efficients are all monomials save one, 

 which is a binomial) was soon afterwards confirmed by an 

 independent calculation. The writer of this notice was 



