535 
1907-8.] Algebra after Hamilton, or Multenions, § 7. 
This shows that f is an invariant. 
Let \ 1 = qi 1 q~ 1 , A 2 = gt 2 g _1 , .... and let p be an independent variable 
fictor given by 
p = x Y i Y + x 2 i 2 + ....= */Ai + y 2 X 2 + . . . . 
[Here x v x 2 , ... . regarded alone are independents ; and y v y 2 ... . so re- 
garded are independents. Of course each set is expressible in terms of q 
and the other set.] 
Let 
V = 3iL) a , v' = 2AJD r 
We have to show that \7 — V'- Now by (18) § 6 
Sdp|v-2 = Sd/°|v / -# 
where q is any function of p. That is 
Sdp|v = Sdp|v' 
for all values of dp. Hence by (10) § 2 v = V- 
The n-tic. — I would here recall the reader’s attention to the equations 
(20) to (27) § 6, where £ (c> and rj ic) first appeared, and would add an obvious 
deduction from (23) § 6 [on the lines of (24) and (26) § 6], namely, 
L (rj?, rj?) = 2L(a?, a£>) = 2L(a?, a?) . . .(19) 
</> satisfies the n - tic 
<p n —h , <f> n ~ 1 + ti'cf> n - 2 - . . . . + = 0 ) 
where h {c) = S ) 
We can scarcely hope, I think, to throw this famous theorem into a 
more compendious form, or one which by (19) can so easily be transformed 
into many other very general forms (to take but one example : put 
ctl = a 1 = flJ a 2 =za 2 = i 2 , etc.), or one which more clearly calls attention to 
the many invariants associated with it. 
To prove (20) we must first enunciate regarding the n - tic what is 
obvious from chapter iv. of Octonions. This part of the chapter is taken 
almost directly from Grassmann’s Ausdehnungslehre. 
h (c) is defined as the coefficient of ( — x) n ~ c in 
S n (</> — x)af(p - x)a 2 .... (</> - x)a n . S^c^cuj ... a n = 1h^ c \ — x) n ~ c . (21) 
where a v a 2 , . . . . a n (our former, § 6, a_ v a_ 2 , .... which we now discard) 
are n independent fictors of the given fictorplex of the n th order. [It is 
not necessary for our present purpose (since it follows by our present 
method) to show that (21) is independent in meaning of the particular set 
a v a 2 , .... a n selected, though this is shown in Octonions.] 
It is then shown that if the roots of Ui (c) ( — x) n ~ c = 0 are a occurring A 
