396 On the Involution and Evolution of Quaternions. 
For the q system of values k — k' = l,2,o...q, the coefficients 
A and B will be real, for the other q 2 — q systems of values 
imaginary; so that there are q quaternion-proper qth. roots of 
a quaternion-proper in Hamilton's sense, and q 2 — q of the sort 
which, by a most regrettable piece of nomenclature, he terms 
bi-quaternions. The real or proper-quaternion values of mi are 
f-ni SI 
1110 (. 
„„(g ± 2to)m +r;T , ( g -l)(g + 2to) l 
q p q y 
fx q meaning the or (when there is an alternative) either real 
value of the qth root of the modulus. 
In the qth. root (or power) of a quaternion m, the form 
Ara + B shows that the vector-part remains constant to an 
ordinary algebraical factor pres ; and we know a priori from 
the geometrical point of view that this ought to be the case. 
When the vector disappears a porism starts into being ; and 
besides the values of the roots given by the general formula, 
there are others involving arbitrary parameters. Babbage's 
famous investigation of the form of the nomographic function 
of — — - of x, which has a periodicity of any given degree q, 
is in fact (surprising as such a statement would have appeared 
to Babbage and Hamilton) one and the same thing as to find 
the qth root of unity under the form of a quaternion ! 
It is but justice to the eminent President of the British 
Association to draw attention to the fact that the substance 
of the results here set forth (although arrived at from an inde- 
pendent and more elevated order of ideas) may be regarded 
as a statement (reduced to the explicit and most simple form) 
of results capable of being extracted from his memoir on the 
Theory of Matrices, Phil. Trans, vol. cxlviii. (1858) (vide 
pp. 32-34, arts. 44-49). 
