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VII. On some Forms of Quadratic Moduli. By Prof. Young *. 

 In a Letter to Prof. De Morgan, fyc. 



My dear Sir, 



I HAVE read with much interest your paper from the 

 Cambridge Transactions, On Triple Algebra. You are 

 aware that I had no access to it till last week, and therefore 

 may not as yet be fully in possession of all your views. It 

 has occurred to me, however, that your system, or rather that 

 particular system upon which you have more especially dwelt, 

 may be arrived at in a manner somewhat more simple, by a 

 process analogous to that which was employed by Euler and 

 Lagrange for finding algebraic functions whose products are 

 similar functions, and which process is nearly the inverse of 

 that adopted in your paper. It is this : since 



(x + ay + o?z){x + /3y + B 2 z) 



=x*+ (a + j8)*y + (a 2 + j8 a )a?* + «j8y* 4- (a 2 £ +^cc)yz + u^z% 



it follows that if a and b are the coefficients of v and v° in that 

 quadratic equation, a 2 — av + b — O, whose roots are « and |3, 

 we shall have 



(x+uy-\-a 2 z)(x + fiy-{-^' 2 z)=x i + axy + (a i -Qb)xz\, ,j s 



+ by* 2 + abyz + Wz 2 J 



and similarly for 



y + a j/ + aV)^ + isy + V) . 



But, as well-known (see Additions to Euler's Algebra), 



(x + ay + a 2 z)(x'-\-ay' + u 2 z')==X + ctY + a 2 Z\ 



(x + Py + p 2 z)(x' + pi/ + p 2 z') = X+pY + e 2 Zj ' ' (2,) 



These products being of the same form as the original factors, 

 it follows that the product of the four factors on the left of (2.) 

 will give a result of the form (1.) ; that is to say, we shall 

 have 



{x 2 + axy {■ (a 2 — 2b)xz + by 2 + abyz + b 2 z" 2 } 



x {x" 2 + axy + (a 2 - 2b)x'z' + by' 2 4- aby'z' + 6V 2 } 

 = X 2 + aXY + (a' 2 -2b)XZ + bY 2 + abYZ+b' 2 Z 2 



(3.) 



J 



Lagrange, in discussing the quadratic forms of reproducing 

 functions, confines himself to those involving only two arbi- 

 trary quantities, x and y ; and when three enter, he considers 

 them exclusively in reference to forms of three dimensions, 

 and so on. From these the inferior forms may no doubt be 



* Communicated by the Author. 



