1899-1900.] Prof. Macfarlane on Hyperbolic Quaternions. 173 
made, x may be equal to y, what then becomes of the modulus ? 
The answer is, the cosine is then — , which shows that the angle 
is infinitely great, and this is the geometrical truth. Suppose 
that the objection is made, x may be less than y, what then 
becomes of the modulus ? The modulus then takes on a form 
appropriate to the conjugate hyperbola, and by the hypothesis the 
angle lies in the conjugate hyperbola. 
The above expression for a spherical quaternion has a resem- 
blance to the Drelistreckung of Professor Klein. But r does not 
mean an expansion and e&v/-i/3 a rotation; the former is a multi- 
plier simply, and the latter a circular angle. The existence of the 
analogous expression re b P, and the application of these expressions 
to develop the trigonometry of surfaces of the second order 
show that his theory of quaternions is inadequate, and the 
sphere of applicability which he assigns them too narrow. 
According to his idea, quaternions will be in place when we wish 
to have a convenient algorithm for the combination of rotations and 
dilatations; the true idea is that quaternions contains the elements 
of the algebra of space. 
In investigating the fundamental principles of hyperboloidal 
trigonometry, the first problem is to find the general expression 
for a spherical versor, when reference is made to the axis of 
revolution. 
Let OA (fig. 2) represent the axis of revolution, and let it be- 
denoted by a. Any versor, POA, passing through the axis of 
revolution, may be denoted by where f3 denotes a unit axis 
perpendicular to a. Similarly AOQ, another versor, passing 
through the axis of revolution, may be denoted by e<V-iy } where y 
denotes a unit axis perpendicular to a. The product versor POQ 1 
is circular, but it will not in general pass through OA ; let it be 
denoted by 
K o w e°Y -1 £ = e & Y ~ i/ 3 e c v / - iy 
= (S + V)(S # + V') 
= SS + SY' + S / Y + YY' 
cos b cos c 4- cos c sin bj _ + cos b sin c J - \y + sin b sin c J _ \/3 jH\y 
= cos b cos c — sin b sin c cos /3y 
+ J - 1 {cos c sin b-/3 + cos b sin c.y - sin b sin c sin /3y./3y}. 
