563 
1907-8.] Algebra after Hamilton, or Multenions, § 10. 
to have given constant values. The general type of integration to be 
considered is sufficiently illustrated by supposing a quadruple integral 
obtained by varying u, v, w, z independently between given limits. 
Any such integral may in the first instance be supposed expressed as 
j du J dv j dw I dzV (u, v, 
w, z 
\ 
/ 
(where V is not necessarily a scalar) and thereafter expressed in equivalent 
increments dp v dp 2 , . ... of p. Let dpjdu = p u . Thus 
dp = p u du + p v dv + p w dw + p z dz 
= d v p + d v p + d w p + d z p 
which is meant to define the independent infinitesimals d u p, etc. 
[d u p = p u du]. Here 
S ipuPvPwPz’dudvdwdz = S A (d u p. d v p. d w p. d z p) 
or dudvdwdz = S± l p u p v p w p z . S ±(d u p. d v p. d w p. d z p ) . 
This shows that in such integrations as I have attempted to describe 
the “ element” of integration may always be taken f(p).S c dp 1 dp 2 .... dp c , or 
more generally as 
cf}Q c dpidp 2 . ... dp c 
where (p is any (scalar, fictor, or the like) linear function of a c th order 
multenion and dp v dp 2 , .... dp c are c independent increments of p. 
The integral (of c integrations) itself may be denoted by J (c) j0c£p[. c) . The 
change of variable from p to or is at once given by (11) and (17) § 8 thus, 
or [(19) § 8] 
Put c = n in 
/7 w= /AiJ} 1 ’* oP ■ 
(15). Thus 
. d o-|f 1 = 4>(^n ) Sdo-^rjP 
dp 
d(r_ 
[(H) §8] 
[(26) § 7] 
[(26) § 6]. 
(15) 
(16) 
Finally putting <p = XS / m l ( ) where X is a scalar function of p and 
xs = (,i 2 we get the usual theorem in the form 
) /xoiw-/ /x 
,J R 
do- 
Z3\da ( y 
