1899-1900.] Prof. Macfarlane on Hyperbolic Quaternions. 177 
OPXQ, which is a sector of the hyperbola having OX for semi- 
major axis, and for semi-minor axis OB which is equal to OA and 
perpendicular to 0 A and OY. This hyperbola is not an equilateral 
hyperbola; PXQ is the curve of intersection of the hyperboloid 
with a plane through the points 0, P, Q. An angle of this hyper- 
bola is specified by the ratio of the sector to half of the rectangle 
formed by OX and OB. Thus a is the ratio of the sector POQ to 
half of the rectangle formed by OX and OB. 
Hence the product versor may be expressed by means of a 
hyberbolic angle a and a hyberbolic axis of the form 
cosh 0’e+ J - 1 sinh O’a, 
where, as before, e denotes a unit axis normal to a, the axis of 
revolution. Let £ denote the above axis ; the actual components 
from which it is constructed are cosh 0‘e and sinh O’a It is not 
of unit length, but it has a unit modulus, The former is 
\/cosh 2 0 + sinh 2 (9, the latter is \/cosh 2 0 - sinh 2 0. 
Hence the product versor may be expressed by 
_ gffl (cosh 0-e+sinh 0-a). 
And to determine these quantities we have the three analogous 
equations 
cosh a = cosh b cosh c + sinh b sinh c cos /3y (1) 
, . sinh b sinh c sin By 
cosh 0 = r— r — 
sinh a 
cosh c sinh b’/3 + cosh b sinh e'y 
sinh a sinh 0. 
As e is of unit length, it may be expressed as cos + sin cf> -k, 
and if i denotes the axis of revolution 
£= cosh 0 (cos cf)j + sin <f>‘k) + J -l sinh O’i. 
The axis £ is evidently a vector to a point in the conjugate hyper- 
boloid of one sheet. 
In the above investigation it is assumed that the magnitude 
of the perpendicular component of the Sinh is necessarily greater 
than the component parallel to the axis of revolution. This means 
that 
cosh 2 c sinh 2 b + cosh 2 b sinh 2 c + 2 cosh b cosh c sinh b sinh c cos fiy 
>sinh 2 b sinh 2 c sin 2 /3y. 
Let sin fiy= 1, cos /3y = 0; then each of the two terms on the 
left is greater than the term on the right of the inequality. Let 
VOL. XXIII. 
M 
