Joi.Y — 0)1 flir Multi-Linear Q/iafcrnion Function. 51 



the whole sixteeti-systevi of such transformations. This appears on 

 expressing a and h in terms of sets of scalar parameters ; and then from 

 (23) we obtain sixteen distinct transformations corresponding to the 

 sixteen products of the scalars of one set by those of the other, 



12. From the equation for a trilinear function 



^af{hcd) = 0. (24) 



it is easy to derive, by permutation and conjugation, scalar equations 

 of the form 



F({ab}, {ed})=0, (25) 



which is combinatorial with respect to a and b, and also with respect to 

 c and d. Thus given a line ah, (25) represents a linear complex ; 

 and in this equation there is a relation between line and complex 

 and complex and line, somewhat analogous to the relations connecting 

 generators of opposite systems of a quadric. 



13. It is possible by suitable permutation to derive from f{ahc) a 

 combinatorial function of a, 1), c; or, in otlier words, a linear function 

 (compare (27)) 



Flahc'] • (26) 



of the symbol of the plane [abc] containing these points. And in like 

 manner from ^af{bcd), we may deduce a scalar combinatorial function 

 of the four points ; but this is simply {ahed) multiplied by a scalar 

 determined by the nature of the function. 



Following out this line, it appears at once that the various per- 

 mutates of a function of the fifth order are not independent. In fact, 

 for the trilinear function, we find the combinatorial function (26), or 

 more fully 



f{ahe) +f{hca) +f{cah) -f{cba) -f{bac) -f{ach) ; (27) 



and similarly from the permutates of a function of the fifth order, we 

 can obtain a combinatorial function of the five quaternions a, h, c. d, c, 

 and the function F (abode). But a combinatorial function of five 

 quaternions is zero; and consequently the Do permutates of the 

 function are connected by one identical relation. In like manner, for 

 functions of higher order, the permutates obtained from any group of 



