Prof. Young on some Forms of Quadratic Moduli. 47 



and, as before, we may, if we please, change the roots of + 1 

 to those of —1, which is equivalent to changing the sign off. 

 I remain, my dear Sir, 



Very faithfully yours, 

 To Professor De Morgan. J. R. Young. 



Belfast, May 2, 1848. 



P.S. At the close of my recent paper in the Irish Transac- 

 tions, I have said, in reference to the expression ax 2 3 + bx 3 y 3 

 + cy 3 % that " no part of one term can be cancelled by a part 

 of another." I should have said, that no part of ax 3 2 can be 

 cancelled by a part of bx 3 y 3 + cy 3 2 ; from which it follows that 

 •r 3 must be divisible by a, and {bx 3 + cy 3 )y 3 by a 2 ; so that y 3 

 must be divisible by a. 



In the paper here referred to, the following eight-square 

 formula has been obtained. It differs from that previously 

 arrived at by Mr. J. T. Graves, only by the introduction of 

 the general coefficients ; that is to say, it differs from Graves's 

 formula only as that of Lagrange differs from the four-square 

 formula of Euler : 



(s/2 + u n + cu n + fcpH + ahea ji + aca j2 + ahy n + az n^ 



x (s 2 + bt 2 + cm 2 -f bcv 2 + abcw 2 + acx 2 + aby 2 + as?) 

 = s" 2 + bt" 2 + cu" 2 + bcv" 2 + abed' 2 + acx" 2 + aby" 2 -f az m . 

 The expressions s", Vb.t", v / c.w",&c, whose squares form 

 the terms of this product, arise from multiplying the factors 



(s?+ \/b.t'+\^c.u'+ ...), {s+Vb.t + Vc.u+ ...) 



in a peculiar way, and not in the manner in which we should 

 proceed if our object were to find the product of those factors: 

 yet all the items in the result severally agree with those which 

 form the common algebraical product, with the exception of 

 the signs. And if from this partial coincidence we were led 

 to inquire what laws of combination, as respects signs, must 

 be assumed for the coefficients Vb, */c, &c, so as to render 

 the coincidence complete, we should be conducted to those 

 which Mr. Graves impressed upon his seven symbols I, m, n, 

 o, i f j, k, in framing his theory of octaves, as explained in a 

 Note, which Sir William R. Hamilton permitted me to add 

 to my paper. I may perhaps be allowed to observe, in refer- 

 ence to the eight-square formula above, that if we makea= + 1, 

 it may be otherwise written thus ; from which several subor- 

 dinate forms, not without interest, may be deduced : 



{ (V 2 ± s' 2 ) + b{t' 2 ± y' 2 ) + c{u' 2 ± x' 2 ) + bc{>d 2 ± w' 2 } 



x { (s 2 ± z 2 ) + b(t 2 ± y 2 ) + c{u 2 ± x 2 ) + bc{v 2 ± to 2 ) } 



= (s" 2 ± «" 2 ) + b(t" 2 ±y m ) + c{u" 2 ± x" 2 ) + bc{v" 2 ± w/' 2 ). 



J. R. Y. 



