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XXXI. Concluding Remarks on a recent Mathematical Contro- 

 versy. By His Honour James Cockle, M.A., of Trinity 

 College, Cambridge, F.R.A.S,F.C.P.S. $c., The Chief Justice 

 of Queensland*. 



AMONG mathematicians there are those who will lend but 

 an academic faith to Mr. Jerrard' s assertion that he has 

 succeeded in rescuing from the class of impossible problems the 

 noted problem of equations. His theory is erroneous, unsup- 

 ported by calculations of his own, and at variance with the results 

 of the calculations of others. Mr. Jerrard may regard article 4 

 of his paper of December 1862 (I refer to the date of the N umber 

 of this Journal in which the paper appeared) as a sufficient 

 answer to me, but I do not so consider it ; and mathematicians 

 will form their own opinion as to whether objections which I 

 have urged against Mr. Jerrard' s analysis are not fatal to his 

 theory. Mr. Cayley's objections Mr. Jerrard only attempts to 

 answer by general observations in articles 1 and 2 of his paper, 

 and by a verbal criticism in article 3, — article 2 consisting in great 

 part of a repetition of a fallacious argument, the use of which leads 

 me, I confess, to the conclusion that Mr. Jerrard has misappre- 

 hended Lagrange's theory of similar functions. Following an ana- 

 logous method to that pursued by Mr. Jerrard, we might dispute 

 the validity of any mathematical proposition whatever on such 

 grounds as these : 



x=x, .'. x— #=(1 — l)a? = 0, 







and all formulae into which x enters are illusory. If Mr. Jerrard 

 were to form a sextic of which the roots should be the fifth 

 powers of those of the sextic in 6 discussed by me, and verified 

 by the independent processes of Mr. Harley, the sextic so formed 

 would (according to Mr. Jerrard, I mean) depend directly ou an 

 Abelian equation, and therefore involve in its solution quadratic 

 and cubic radicals only. If Mr. Jerrard should refuse to test 

 his theory by forming and trying to solve the latter sextic, there 

 may be no impropriety in suggesting the following test. Assu- 

 ming the coefficients at his pleasure, let Mr. Jerrard construct 

 an irreducible quintic into a root of which an irreducible cubic 

 surd shall enter. That the former test will not have for its 



* Communicated by the Author. 



