1907-8.] Algebra after Hamilton, or Multenions. § 10. 559 
The form £q + rjr is especially simple for some purposes. Thus 
J)- 1 = £qr l + rjr-\ p\p 2 = £q x q 2 + rjr p’ 2 , 
or more generally if f(p v p 2 .... ) is a given function of p v p 2 , .... 
/(p 1 ....) = ^/fe....) + ^/(r 1 ....) . , . (41) 
The above will, I think, bear out the contention that the concept of 
the continent multiplex is a mischievous excrescence * on the present 
method ; the reason why I was initially misled into contemplating it will 
be obvious enough to quaternionists. 
From (34) we have, when q and r are of even order, 
C(q + n 0 r)=q + r 
and therefore (1 - C)(q + ST 0 r) = ( ZJ 0 - 1)?* = - 2 rjr. 
Now C p=p 0 and (1 — C )p—p 1 are two parts of p, such that 
P=Po + P 1 , Cp 0 = Po’ CPi^ 0 .... (42) 
Hence another convenient bi-form of p is 
p = q + y,r (43) 
The complementary replacement (C) of q is q. 
10. Differentiation, Integration, Jacobians. — If the reader thinks 
that 
ought to mean qr 1 , even when q and r are differentials, and therefore 
demurs to the meaning about to be given of 
he may bring into use the old significance of the colon, and may mentally 
read dor : dp or 
da- 
dp 
wherever dcr/dp occurs below. 
If y v y 2 , . . . . , y n are n scalar functions of n scalar variables x v x 2 , ... . x n , 
in our present subject we should take account of this by saying that the 
fictor cr is a function of the fictor p, where 
p = ^q + . . . .+X n > n ( 
^ = V\h + ••.. + y n ' n f 
* Mischievous because it hides some of the inherent simplicities of the method. 
