529 
1907-8.] Algebra after Hamilton, or Mnltenions, § 6. 
The following obvious transformations enable us to write down a 
number of alternative forms of (1), (2), (3), 
s n qr c = S n q n _ c r = S n q n _ c r c ) 
S qr c = S q c r = S q c r c » 
These are analogous to the quaternion formulse 
S.qYr = S.Yqr=SVqVr. 
We will now suppose that a_ x , a_ 2 .... a_ n are independent and will 
use (2). Put 
af- c, S;W- c ) = Ka? (7)* 
The expression on the left is clearly of the c th order, so that the suffix 
of a ( c is justified. The index dash (c) will be justified below. The reader 
can ignore it for the present. [I have had much hesitation whether to put 
d ( c or K a ( c on the right of (7). The formulae of this section are more 
complicated with the K than they would be without, but there is more 
symmetry between a (c) and d {c) ; and the translation of our results into 
ordinary .algebra is simpler. On the whole, perhaps the plan adopted is 
better than the alternative.] 
From (7) and (2) we have 
Sa?Kd?=l (8) 
q = S 2 a?Sa< c) Kg (9) 
q c = M c) ^Kq (10) 
the summation in the last, of course, only referring to the n C c products 
which contain c fictors. 
Putting c equal to 1 in (7) there are n fictor values of d { c c) depending on 
which of «_ 1} .... a_ n is omitted. These we denote by a|| a_ 2 , .... and 
choose them so that (in harmony with (8)) we have 
Sa_ 1 Ka_ 1 = Sa_ 2 Ka_ 2 = . ... = 1 . . . . (11) 
that is, we put 
Ka_ x = S n _ 1 a_ 2 a_ 3 .... a_ n .8~ 1 a_ 1 a_ 2 .... «_,| \ 
-Kft_2 8 n _ x Q,_ x Ct_3 .... <X_ n . Q-_ X Ct_2 .... Q_7 i V • • (12) 
In (12) we may interchange the set a_ x , a_ 2> .... a_„ with the set 
d_ i, a_ 2 , .... a_ n that is, the two sets are symmetrically related. This can be 
proved directly from § 5, but it is more easily deduced from the particular 
* There is another and perhaps simpler method of dealing with what follows, by means 
of Unities. Let <f> be given by a l5 a 2 , or the latter by the former by the equation 
= a_ 1 Spi 1 -1 + a_ 2 Spt 2 _1 + .... 
so that a_ x =<pi 1 etc. (p~ l is not infinite. Then o_ 1 = 0 1_1 i 1 , a_ 2 = 0 1_1 t 2 etc., and in the 
notation of § 8 below aJ c) = <P c t (c) etc., ,a c (c, = 0 c 1_1 i (c) etc . — [Note added April 1908.] 
vol. xxviii. 34 
