4 EE. B. Wilson, 
becomes necessary to redefine the method of evaluating the product. 
Thus arises the theory of regressive multiplication in which the 
class of the product is # + /— um instead of k+ 7/2. This product 
is also treated in detail in the references just cited. 
It should be noted, however, that the theory of regressive multiplica- 
tion, which is usually based upon the theory of supplements in the 
Grassmannian sense, was treated in an entirely different manner by 
Gibbs. His point of view and method of procedure were outlined in his 
address on multiple algebra already cited: but as that presentation is 
extremely brief, it may be well to recapitulate his method in some 
detail. Let «, 8, y, ... be any number of elements of the first class. 
Consider the product of two factors (the cross introduced in (2) may be 
omitted for brevity in writing) 
(@pyod. 2.) (...Apya) 
each of which contains not more than x elements, say # and / respect- 
ively, but which together contain more than 2 elements. The product 
of two such factors is called regressive when computed by either of the 
following rules: 
1°. From the second factor (...4uva) take enough, that is, x — &, 
of the remoter (last) elements to form a total of x with all the elements 
of the first factor (« Byd...), thus obtaining a scalar (a product of the 
nth class) to serve as coefficient to the remaining elements of the second 
factor. Do this for every permutation of the Z elements in the second 
factor (...4uyz2) which may be necessary to bring every combination 
(not permutation) of 2 — % of them once and only once to the end of the 
factor, and add the results thus obtained with the positive or negative 
sign according as the number of simple transpositions of the 7 elements 
in any permutation is even or odd. 
20, From the first factor take enough, that is, x — Z, of the remoter 
(first) elements to form a total of ~ with all the elements of the second 
factor, thus obtaining a scalar to serve as coefficient to the remaining 
elements of the first factor. Do this ... and so on, as before. 
Thus it 7 = 4, the following expansions of regressive products are in 
accordance with the rules just stated. 
u(pydé) =(aydé) p—(ades)y+(aesy)d—(eByd)eé, 
(a By) (de) =(a@Bye\d—(aPyd)e=(apde)y+ (By de) at (y ade) 8B, 
(aBy)\GelC)=(apyode+(apyesCd+(apydjel=(adel) py 
+ (sdelC)yat(ydetap. 
It may be remarked that in the first line, the product on the left of the 
sign of equality is already expanded as far as possible by the second 
of the rules. Furthermore, if x had been 5, the last product would 
have been 
(aby) Gel) =(apyeddt+t(epyde)Et(apyld)e=(aspdeljy 
+ (Byde0Hat(yadel)p. 
It is hardly necessary to note that the signs in the expansions may all 
be taken positive by properly arranging the permutations on the letters. 
