202 G. PLARR ON THE ELEMENTARY PRINCIPLES OF QUATERNIONS. 
We may remark the formulee— 
See XP les 4) 
re Gia ) 
K'a~p) =p 
Supplementary Note. 
With the rules on the four operations, addition, &e.... division, at our dis- 
posal, we are enabled to reduce to a quaternion any algebraical function of vectors. 
The problem of this reduction, with its rules of abridgment for the separate 
formation of the scalar of the function, and of the vector of the function, con- 
stitutes a distinct chapter in the theory of quaternions, into the details of which 
we do not intend here to enter. 
We confine ourselves to the indication of two results, which relate to the 
general question. Let a, a, &c., a,, represent ” vectors, generally different 
from one another, and let a, represent one of them. We may conceive the 
product of » — 1 factors, for 4 = 1, 2,...n, 
Pr = An4+1 A, +2 SP Or enna 
where the index exceeds , but where it is to be reduced to be not greater 
than n, by the supposition 
Ag = Ag—n = gon ’ 
so that it may be positive, and not greater than » in each of the factors. 
This conceived, let us form the sum of terms 
3h (—1)'*! a Spr, 
the sign > indicating by its limits that / is to receive the m values 1, 2, 3.... 
n —1, n, upper limit included. 
This sum expresses two different results, according as 7 is an even or an odd 
number. 
When 7 is an even number the sum is equal to zero. When nv is an odd 
number, the sum gives the expression of 
V (a; a nung An) s 
as a linear function of the single factors, respectively multiplied by a scalar. 
It seems impossible to express in a similar form the vector of the product 
of an even number of vector factors. The expressions of such a vector by a 
linear function of the vectors of the 5 n(n —1) combinations 2 by 2 of the 
factors is possible, but of little simplicity when m exceeds 4. 

