double, treble, and other Systems of Algebraic Equations. 433 



so that a + /3 4- y for this case becomes greater than a-{-b + c, 

 and the method falls to the ground. 



In fact, I have discovered a theorem which lets me knovv' this, 

 a priori, a law which serves as a staff to guide my feet from 

 falling into error in devising linear methods of solution, and 

 the importance of which all candid judges who have studied 

 the general theory of elimination cannot fail to recognize. To 

 wit, if Xi Xo X3 . . . . X^ be {n) integer complete polynomial 

 functions of n letters o-j a-j . . . ^',1, and severally of the degree 



b^bc^b.^ bj^; then it is always possible to satisfy the 



identity, 



Pi.X, + P,.X, + P3.X3+ .... + P,„.X, 

 = F . .ri«i . x./; . xf3 . . . a-^^n 



if a, + «.^ + «3 + + «« ^^ equal to or greater than 



6, + ^2 + ^3 + + ^,j — ?« + 1, but otherwise «o^*. 



This again is founded immediately upon a simple proposition, 

 of which I have obtained a very interesting and instructive de- 

 monstration, shortly to appear, and which may be enumerated 

 thus : " The number of augmentees of the same degree that can 

 be formed, linearly independent of one another, out of any num- 

 ber of polynomial functions of as many variables, may be either 

 equal to or less than the number of distinct arguments contained 

 in such augmentees, but never greater. The latter "will be the 

 case when the index of the augmentees diminished by unity is less 

 than the sum of the indices of the original unaugmented poly- 

 nomials each so diminished; the former, vohen the aforesaid index 

 is equal, to or greater than the (foresaid sum." 



To return to the particular case of finding X, Y, Z to sa- 

 tisfy X . IT + Y . V + Z . W = F . .r?' .y . ::'. 



This has been already done according to the first method; 

 if we employ the second method of elimination we shall have 

 f+g + h = 2n-2; 



* Hence it is apparent, tiiat in applying the method of multipliers, a curi- 

 ous and important distinction exists between tlie cases of there being two 

 equations, and there being a greater number to eliminate from : for in tiie 

 first case the element of arbitrariness neeils never to appear; in the tatter 

 it cannot possibly be excluded from appearing in the multipliers. 



Tiiis will explain how it comes to pass that the method of the text may 

 be emjiloyed to give various solutions of the X . U + Y . V + Z . \V 

 = F..r''.y' .z'; thus not only can (p), (r/) and (r) be variously made up 

 of (/+ a), (g + I/), {/i + c), but also II lcc,ft, y) when two of the indices 

 (a, /3 suppose) are each not greater than the assigned greatest values a, h 

 may be made to figuri; indiHercntly either imder the form 



Ti.U + ^.V + V.W , x'.U + (C4.V + V'.\V 



E. or that ot . 



X* x<^ 



Phil. Mas. S. 3. Vol. IS. No. 1 19. June 1811. 2 F 



