8 EE. B. Wilson, 
(4’) (PLA)<M=P(AxM=(4*MT 
where 4 >I” is a scalar, and the result is therefore an element of 
the same class as that which was multiplied by the dyad. 
3. Leciprocal sets of elements.—The theory of reciprocal sets is 
fundamental to the entire treatment of multiple algebra as here 
given. To a large extent it obviates the necessity of discussing the 
theory of supplements in the Grassmannian sense. In fact, by his 
definition and treatment of regressive multiplication and by his theory 
of reciprocal sets, Gibbs entirely avoided the supplements .in his 
course. Before proceeding, however, to the reciprocal sets, it will 
be well to introduce once for all the change of notation already 
adopted in the discussion of regressive multiplication. The sign of 
the combinatorial product, the cross ><, occurs so frequently as to 
render the formulas too bulky. I shall therefore write 
IA instead of =< A 
for the combinatorial product. This necessitates a different notation 
for the dyadic product, and I shall write for this product 
I'| A instead of IJ, 
where it should be noted that the vertical bar has no relation to 
the Erganzung of Grassmann. This is in entire accord with Gibbs’s 
procedure in his lectures; the change is made purely for convenience. 
The reason that this notation was not adopted from the start was to 
emphasize the fact that the dyadic product was fundamental and 
the combinatory product merely a function of it. 
Let there be given w independent primary vectors or elements 

Gy; 0b; s-2 3 OR Gy Oy 1 yy == 0: 
Form the 7 expressions 
Cit Bio +2. On Oy Oo... Car : 
(5) ec, = + a = Sere 
Qity Wito... Om Oy Ay... Gia G% 
The # quantities «’;, @’s,... a’, thus obtained are elements of the 
(w—1)st class. Taken as a set, they are called the reciprocal set 
to the 7 elements @, @,..., @n. For brevity a‘; is sometimes called 
the reciprocal of a, From the definition (5) of the reciprocal set 
it appears that the elements and their reciprocals satisfy the equations 
(6) ey; a; = 4, a7 a; = 0, a = 7. 
By the laws of regressive multiplication it follows that the 
n(n—1)/2 elements a‘; a’; of the (z—2)nd class and the equal number 
of elements «; a of the second class satisfy the equations 
(6') aecaj;aag=1, erajya,na=O0, kk and / not both 
equal to z and ». 
Similar equations are satisfied by the elements «‘; a; a‘; of the 
