doiibley treble, and other Systems of Algebraic Equatiotis. 429 



that is to say, becomes a linear function of the augmentatives, 



and therefore if combined with them in the process of linear 



elimination would give rise to the identity = 0. 



Hence we must reject all such secondary derivatives as have 



zero for one of the indices of derivation. But all others, it 



may be shown, will be linearly independent of one another, 



and of the augmentees previously found. Hence, besides 



^ n .n + 1 . „ p , , 



3 . equations ot augment ot the degree 2 ?i — 1, we 



shall have of the same degree so many equations of derivation 



as thei'e are ways of stowing away between three lockers 



(71 + 1) things, under the condition that no locker shall ever 



1 1 r , • M . (« — 1 ) „ 

 be left empty, i. e. — ^-- ^.* 



Thus, then, in all we have n . — - — -f 3 . ~ — — — 



2 n (2 71 + 1^ 

 = — ^^ — 2 equations, which is exactly equal to the num- 

 ber of arguments to be eliminated. Hence the final derivee 

 can be obtained by the usual explicit rule of permutation, and 

 moreover will be its lowest fo7-m, for it will contain in each 



71 . {71 -^ \\ „ , , . , 



term — ^— prehxes belongmg to the augmentatives of U, 



and a like number pertaining to those of V and of U, as well 

 as 71 . — - — belonging to the secondary derivatives, each pre- 

 fix in any one of which is triliteral, containing a prefix drawn 

 out of those belonging to each of the proposees. 



Thus every member containing 7i . -~ \- n . , i. e. 



71? of the original prefixes belonging to U, V, W, singly and 

 respectively, the final derivee evolved by this process will be 

 in its lowest terras ; as was to be proved. 



Case A. — Indices all equal. 

 Method 2. 

 It is remarkable that we may vary the method just given by 

 making /• + 7-'+ r'' = ?j — 2, s+ s'+ s'' = ?z — 2, t + i'+i'' = n — 2. 

 The augmentatives will thus be of the degree 2 ?/ — 2. 



Furthermore, we must make a + /3 + y = ?i + 2. It will 

 still be possible to satisfy by integer multipliers the equations 

 U = a:" . F + / . F' + ^'' . F", 

 V =x«.G -f /.G'+ z"^ .G", 

 W = x". H + /. H'+ ^''. H". 

 • Vide page 42G for the Latin version. 



