1899-1900.] Prof. Macfarlane on Hyperbolic Quaternions. 171 
(3 and y. Similarly for any two imaginary axes J -1/3 and J - ly 
( n/- l/3)( n/- iy)= — cos fly - sin /Sy J - l/3y. 
I proceed now to apply these principles to the investigation 
of the fundamental theorems of hyperboloidal trigonometry. I 
shall consider only the hyperboloid of equal axes, but the results 
can easily be extended to the general hyperboloid. 
On account of the symmetry of the sphere with respect to its 
centre, spherical quaternions are independent of rectangular axes. 
It is otherwise with hyperboloidal quaternions, for the equilateral 
hyperboloid has an axis of revolution. In order to treat of 
trigonometry on the hyperboloid, it is necessary first to treat the 
trigonometry of the sphere with reference to the same axis of 
revolution. In the figure (fig. 1) OA is the axis of revolution, 
and the surfaces considered are those generated by the circle and 
by the equilateral hyperbolas. From this point of view the circle 
appears as consisting of a real part PQ corresponding to the real 
hyperbola P'Q', and an imaginary part QR corresponding to the 
imaginary hyperbola Q'R'. Consequently the sphere appears 
broken up into a double sheet traced out by PQ and RS, and 
a single sheet traced out by QR. 
The algebraic expression for a circular angle is e b ^~ l . As the 
axis of the plane is not specified, the denotation of the expression 
is necessarily limited to angles in a constant plane. Let (3 he 
introduced to denote the axis, then eV-i/3 is the proper expression 
for an angle in any plane. We have 
e W-ip=-.l + b J -l{3 + 
(Kt-M 2 , (bj-ipy 
9 t 
3! 
+ . 
Let the principle of reduction be introduced, which reduces 
( J — l/3) 2 = - 1 then the right hand member becomes 
b 2 b 3 
1 + bj — 1(3 ~ o - j ~ 3 ~ |\/ “ 1 ft F etc. 
~ 1 ~Y\ + 3 1 “ 
+ { b -rA rr 
= SUg + VUg 
= cos b + sin b ( J - 1 f3 r 
