571 
1907-8.] Algebra after Hamilton, or Multenions, § 11. 
Hence / ft fa) — £• Also from (20) and (21) 
£fa — a ) = — ^o- 
Hence £ and £ 0 satisfy (18). 
The following is easily proved, and is required below, 
£ of /fa) = £, £<> of /fa) =/(<* + £o) -/(«) I 
[/(« + 4 ) -/(«)? =0 J 
provided f(a),f(b), .... are all different from one another. Here “ (• of 
/fa) ” means the particular £ which corresponds to the root /(a) of /fa), and 
similarly, of course, for £ 0 . 
I have thought the above the best introduction to the standard analysis 
of cp implied by (21), if only because it enables us by the simple rules 
implied by (17), (18) to determine £ f 0 , etc., when either the minimum 
degree identity or the Grassmann identity is known for <p. But the 
definitions of £ etc. might, instead, have been made to depend on the 
fundamental Grassmann theorem (22) of" § 7. We proceed to show with 
the notation of that equation : if p is any fictor 
$p = A/jpjAj + A 2 SpjA 2 +....+ A A Sp|A A 
ioP = V S /°'A2 + t.- • • • + A A _ 1 SpjA A 
where the bar refers to the set of independent fictors A 1 , A 2 , . . . , pq, 
/j, 2 , . . . . (24) is, equivalent to 
= A 1 , £A 2 = A 2 , . . . ., = • • • • = 0 ) (^5) 
To prove these equivalent sets (24) and (25), first note that £ kills every 
/x, v, . . . , rj kills every A, v, . . . , because of the factor fa — 6) ,l+ in £ and 
the like. Thus putting 
pz=tj p + r)p+ .... 
= A + /A + . . . . 
then A = £p is a fictor in the fictorplex (A) [and may be called the com- 
ponent of p in (A) with reference to (A), fa), . . . ], and similarly for \x = r\p 
etc. It also follows that 
fa — «)A = fa — $)£A = £qA = £ 0 p. 
The second of (24) now follows from the fact that (^ — 0 )^ = 0, 
fa — <x)A 2 = A/, etc. 
Equation (22) has useful applications to many other cases than when 
/ is an integral function with a finite number of terms. Thus we may 
clearly put for f(x) any one of the following direct algebraic functions 
(supposed defined by convergent integral expansions), 
e x , cosh x, sinh x, cos x, sin x. 
