the equation x^ -^y^ -\- kz^ — Mxyz, 8^c. 469 



system of values of x,y, z satisfying the proposed equation, 

 in which z is even, is comprised under the form 



and every possible system of such values, in which z is odd, 

 is comprised under the form 



^^f\.4,<p\-i.yl, . . . vI/(p~i.P: 



the quantities n^ Wg • • • ^h being of course all independent 

 of one another, and unlimited in number and value. 



Thus then we may be said to have the general solution of 

 the given equation in the same sense as an arbitrary sum of 

 terms, each of a certain form, is in certain cases accepted as 

 the complete solution of a partial differential equation. 



As regards the value of the symbols ^ and (p, <p indicates the 

 process by which a, b, c becomes transformed into «, /3, 7, the 

 relations between the two sets of elements being contained in 

 the following equations : 



a' = fl3 V = l^ c' = Ac3 



/3 = a' b''' + b' c'^ + c'a'^-SaW 

 y=abc{a'^ + b'^ + c'^-a'b'-a'c' - b'c'}. 



Next, as to the effect of the Duplex symbol rj/. Let eg i 

 be the elements of the Primitive system P: » being the value 

 of z and e, g of x and j/ taken in either mode of combination, 

 each with each, which satisfy the proposed equation 



^ + j/^ + Az^ = Mxyz, 



Let /, ntj n represent any system S, 

 A, ft,, V represent any system 4/(S), 

 \I/S has two values, which we may denote by vJ/'S, '\{/S re- 

 spectively, and accentuating the elements X, fx,, v accordingly 

 to correspond, we shall have 



>J=3ffm{gl—em) + S\m{U—en) — M{gd^ - eHm) 

 ljJ=3Am{i7n —gl) -r Sel{ein —gl) — M{eim^—gHm) 

 v' = 3el{en — d) + 3g7n(gn — nn) — M {egn^— iHm) ; 



we have then 



and in like manner 



'\I/ S being derived from '^ S by the mere interchange of e 

 and g one with the other. 



I have stated that every possible solution of the proposed 

 equation comes under one or the other of the orders, infinite 



