551 
1907-8.] Algebra after Hamilton, or Multenions, § 8. 
we shall 
permit 
. ( 12 ) 
7?, • 
• (13) 
. (14) 
. (15) 
a?>, a? for 
I?- 
(16) 
{< p } is obviously an ordinal multilinity for which { 0 } / c = {<t>'} c - We shall 
therefore denote {(p} c , {<p}' c by <p c , (p' c respectively, though we shall permit 
ourselves to return to the original notation if we please. 
We clearly have 
{<£} = <£o + <£i + <£ 2 + ; ■ • ■ + <£» 
where <f> 0 q = bq, faq = ^rjbqly], . . . . , <f> c q = 
<t>iP = <l>P . ' . 
By § 6 and (17) § 7 transform (11) by substituting a? for ^j, c) , * 7 A 
Thus 
Wq=^M \ 
^ = S(^a)WSg|^> j 
In (16) put q = a ( c\ and to fix the ideas, consider the particular case 
c = 4. Thus 
AA C =(^a)? I (17) 
<j6 4 S^cqagcq = S 4 <^a 1 ^>a 2 ^)a3^)a 4 .1 
by ( 8 ) and (4) § 6 . This equation, which is obviously true when oq, a 2 ,, 
etc., are not independent, justifies the statement that { 0 } is the strain 
operator for space of n dimensions. 
It follows from (17) that 
{W} = MW ( 18 ) 
for i/q</> 4 S 4 a 1 a 2 a 3 a 4 = i^ 4 S 4 ^>a 1 (^a 2 ^)a 3 <^a 4 = S^c/jct] i/f</>a 2 i//<£a 3 ^<£a 4 . 
Summing for all such sets of four fictors out of n given fictors we get 
»A 4 </>4 = W) 4> 
and (18) now follows from ( 10 ). 
Putting \jr — (p~ l in (18) we have 
Wb^HUHi 
or {</>}-!= {^} ( 19 ) 
From these we clearly have 
m=w a (2o> 
for all integral values of a, positive or negative ; and 
{/m=/w ( 21 ) 
when f(x) is a given rational integral function of x. 
An interesting generalisation of S — S /3(p'a is the following, where 
we might have c in place of 4. 
S. S 4 a 1 a 2 a 3 a 4 S 4 ^>a 1 c^a 2 ^)a 3 <^)a 4 
— S. S 4 a 1 a 2 a 3 a 4 <^) 4 & 4 a 1 a 2 a 3 a 4 
= S.(c/) 4 , S 4 a 1 a 2 a 3 a 4 .S 4 a 1 a 2 a 3 a 4 
= S.S 4 <£ cq^ a 2 ^> , a 3 A a 4S 4 a 1 a 2 a 3 a 4 
. ( 22 ) 
