436 Mr. J. Cockle on Quadruple Algebra, 



two expressions of the form A, the modulus of the product is 

 the product of the moduli of the factors. It is for the purpose 

 of analogy and of making the modulus positive, in all cases, 

 that I have given a quadratic form to the formula employed in 

 expressing the cotessarine modulus. 



On Equations of the Tiftli Degree. 

 Whether Mr. Jerrard has succeeded in pointing out a 

 method of solving the algebraic equation of the fifth degree 

 or not, his investigations at pp. 545-574 of vol. xxvi. (continued 

 at p. 63 of vol. xxviii.) of the present Series of this Journal 

 must ever be a subject of interest, and form an essential part 

 of the theory of such equations. There are, however, one or 

 two portions of his papers which seem to me involved in doubt 

 and difficulty — difficulty which, in one case, he has himself 

 adverted to and endeavoured to explain. Mr. Jerrard will 

 pardon me if, with great hesitation, I venture to intimate an 

 opinion that the position taken by him in his note [|] to p. 572 

 of vol. xxvi. is untenable. By way of example, suppose that 

 the square root of a^ + /^ is the function to be expanded. The 

 general form of the expansion* is, — 



A Series of converging or diverging terms ^/m5 a Remainder. 



Now, when the series is convergent, the remainder may, in 

 all cases where numerical value is the subject of inquiry, be 

 entirely neglected ; but it does not the less constitute an essen- 

 tial part of the symbolic expansion. Hence I conceive that, 

 in considering the expansion under a purely symbolic point of 

 view, even the convergent development must be regarded as 

 incomplete without the remainder, and so placed on the same 

 footing as the divergent one. Considered thus, the convergent 

 and divergent developments are deducible, the one from the 

 other, by an interchange of a? and //, and each admits of that 

 interchange without alteration o^ symbolic value. And I think 

 that we necessarily obtain an expression which admits of such 

 interchange — at least in all cases where a strictly symbolical 

 expansion is required ; and, if I rightly understand Mr. Jer- 

 rard's argument, it is to such expansions that his remarks 

 apply. 



But, admitting for a moment that the convergent series with 

 the remainder neglected is &. symbolic expansion of the function, 



• I have elsewhere (in the course of my HorcB AlgebraiccB, Mechanics' 

 Magazine, vol. xlvii. p. 150) suggested contraction as a term to denote the 

 inverse of expansion. Would it be advisable to confine the terms expan- 

 sion and contraction to symbolic operation, and to use the terms involution 

 jind evolution exclusively in reference to arithmetic or quasi-arithmetic 

 operations, including them both under the cominon name volution ? 



