178 Proceedings of Royal Society of Edinburgh. [sess. 
sin J3y = 0 and cos py = - 1, then the above expression reduces to 
the well known inequality a 2 + b 2 > 2 ab. Hence the terms on the 
left are always greater than the term on the right. 
In the case when the two versors are equal, we can verify that 
it is the line of intersection of the central plane with the equi- 
lateral hyperboloid which is indicated by the product of the 
versors. 
As the two versors are equal they might be denoted by e b P and 
e b y. Let cosh b — x, sinh b = y. Then according to the theorem 
e b P e b y = x 2 + y 2 cos Py + xy ({3 + y) + J - 1 y 2 sin py a 
As (fig. 6), OB the semi-transverse axis of the hyperbola PXQ 
is 1, HQ represents the sinh of half of the product angle. How 
by the geometry of the construction 
HQ 
ob =:W 2 r l + 2 r !cos ft' 
y 
= + coa Py- 
. OH x 
Agam OX - cosh 6 
_ x J (x 2 + y 2 cos /3y ) 2 - 1 
Jx 2 y 2 2 (1 + cos /3y) 
= J 1+ y(l+COS /Jy). 
Now cosh 2XOQ = (cosh XOQ) 2 + (sinh XOQ) 2 
/N QV . /ON\* 
\OB/ \OX) 
2 2 
= y (1 + COS Py) + 1 + y( 1 + cos Py) 
= 1 + y 2 + y 2 cos Py 
= x 2 + y 2 cos py 
which agrees with the above theorem. 
We have seen that the general spherical versor is denoted by 
w here 
£= - sin O' a + cos O'e, 
a denoting the axis of revolution and e an axis in the perpen- 
dicular plane. Similarly a general versor for the equilateral 
hyperboloid of two sheets is denoted by e a %, where 
£= J — 1 sinh O' a + cosh 0-e, 
