Double Products and Strains in Hyperspace. 47 
(91) £27 =Z((@B)ala + (aa) Bp\eB— (a 8) a|B — (a8) B| «|, 
where the summation covers all pairs of antecedents «a, 8, y,... 
of 2. The form of this expression evidently bears out the state- 
ment that the expression is independent of the manner in which 2 
is written: for if @ is replaced by a+ a,, the sum of the terms 
due to a@,8|aP8 and a,8\aB is the same as those due to af| «a, and 
so is it for similar changes in any of the elements that enter into &. 
To pass on to 3() consider the typical product 
oy Gee a| jay ap al 
(aByAuo) (AwoaBby) =o 7B “up Bg | Ba Bu B | 0 
PAP ee ay | LIRA! FU Oy 
The coefficient of the dyad aa is 
ies atl Baal 
8 W012 Bel ae ay aw a)—BrUulew ey 
If this be summed over all pairs of different antecedents of /, the 
result is the term (@y8y) aa of ZS). In like manner the coef- 
ficients of @|8, ay, Bly, ... may be found. In fact the rule for 
writing down the desired term is merely to pick out the two ele- 
ments from the set «fyaPy, leaving the other four to form the 
scalar coefficients with the proper sign (which may be determined 
according to the rule for expanding the determinants above) — thus 
—(ayep)yiB, +(aBes)yi7y, —@Byay)sle 
The extension to Z\) is now immediate. There is no need of going 
into the details. 
If 2 is expressed in matricular form as 
2Q= Saijal aj, 2.=$ 2 ai5j Aim a a| ajA'm, . 
it is possible to calculate 3), 3, ... also in matricular form. 
From (9) 
22) = & Saiz aim((@ ma) a| aj + (a7 a) at| am — (a5 a1) a | om 
—(@m a) a| ¢;] 
= 3 ij Aim (em a) a | &j — Vij Aim (a; ar) | am 
It should be noted that 7== / and 7/=— m, and that terms in the first 
sum vanish unless /—= m, in which case @, @—1: and similarly 
in the second sum 7—/ Hence the coefficient of any dyad 
a | ej in Z() is 
(45+ ail 4 a 1.41) aj | aj 
where the accent on the & indicates that those terms for which the 
variable index is equal to either of the fixed indices are to be 
omitted. These sums may be written in the form 
et) za +2 pecaae Wr aj. (¢—2) ¢—7)=|=9. 
Eject aj ai . 
