Double Products and Strains in Hyperspace. 2 
“I 
algorithm and carry the division up to and including the power 
Prp—-! in the quotient. Then 
I Saini ek 
(46) ALL BP pe AIT BE +. ee 1 
pr P(P) 
+ AIF BES...+ HE 
where P(#) is a polynomial of degree m—p—1 in ¥ Set 
Pew tg (AI - Be -.. atmo) (A+ BP... Ee Pe) 
= [_Pp P(P). 
In like manner compute J;, /;, ... corresponding to the values 4, 
Eis 1% 
The dyadic J, does not contain #? as a factor; for it is /— Pr P(#P), 
It does, however, contain (¥—d6 /)7 (6—c 1)" ..., which represents 
the other factors of the equation of least degree. Hence Jy ¥? 
contains the equation of least degree and vanishes. Hence 
Ly? = Ig [I—2? P(P)| = La—h, PP P(P) = Ih, 
The product J, 4, contains in /, all the factors of the equation of 
least degree except (#—a/)? and in /, it contains those. Hence 
fa — 0; Thus 
(48) de =e Ay Sh ho? = Les 
i Ol le — Ol ge Oras, 
Let 2 be the sum 
(49) Oe Fe ao! Pe ae nae 
It follows that 

Von) 
2=22 or (Q—/)2=0. 
This expression is a polynomial in @ and is equal to zero. It must 
contain the equation of least degree as shown in article 6. But 2 
contains no factor of this equation, because any factor such as 
#@—aT is contained in all the /’s except J, Hence all the factors 
of the equation of least degree must be contained in Q—/. As 
Q—T is of degree m—1 in @, which is less than that of the equa- 
tion of least degree. the only possibility is 
(50) Q2-IT=0 and 2=]. 
Suppose J,, /, /.,... written as the sum of the fewest possible 
number of dyads, so that 
(51) a= ajatel@tylyt+..., 
h=AjrAtuluetorviyt+..., 
Then if /,2 be compared with Jj, 
f= (ea) ale (ap) e|P+@y)aly+... 
+ (88) 8\8+ (Ba) Blat+(B@y) Blyt+... 

