1899-1900.] Prof. Macfarlane on Hyperbolic Quaternions. 179 
a and e denoting the same kind of axes as before. This leads 
us to the consideration of hyperboloidal axes. Let £ x denote a 
radius to the double sheet (fig. 7) ; 
= cosh 0 'a-\- J - 1 sinh 0 'e. 
The length of £ x is 
ij cosh 2 0 + sinh 2 0 
but its modulus is J cosh 2 0 - sinh 2 0 , which is 1. Let £ 2 denote a 
radius to the single sheet ; 
i 2 = ^/ - 1 sinh O' a + cosh #*e. 
The corresponding axes for the unit sphere are 
cos 0 'a + sin 6 ‘e 
and £ 2 ~ — s ^ n + cos $’ e - 
Just as a spherical vector is expressed by rj-lg, so a hyper- 
boloidal vector is expressed by r£, where r denotes the modulus 
and £the axis. The principal difference is that in the case of the 
sphere £ is of constant length, whereas in the case of the hyper- 
boloid the length of the axis depends on its position relative to the 
axis of revolution. 
Consider now a general triangle on the hyperboloid of two 
sheets (fig. 8). Let the axes to the three points be denoted by 
£=cosh 
0’a + J - 1 sinh 
6 -ji 
77 = cosh 
O' 'a + J - 1 sinh 
6 'y 
£ = cosh 
0"‘a+ J - 1 sinh 0"‘8. 
Then £77: 
= co^h 0 cosh O’ - 
sinh 0 sinh O' cos /3y 
a) 
l 
0 
0 
GO 
t3" 
sinh O’’ ay - sinh 
0 cosh O’’ /3a 
(2) 
-V-i 
sinh 0 sinh O' sin 
(3ya 
(3) 
Hence cosh £77 = (1) 
and Sinh £77 = (2) + (3). 
We have proved that the length of (3) is always less than the 
length of (2) ; hence £77 has the form 
sinh <b’a + J - 1 cosh 
And the same is true for rjt, and ££. The central section is always 
hyperbolic. 
Now ft=(^)(^)- 
Therefore cosh ££=cosh £77 cosh 77^ + cosh (Sinh £77 Sinh 77^) and 
Sinh ££ = cos 77^ Sinh £77 + cosh £77 Sinh £77 
+ Sinh {Sinh £77 Sinh 77^}. 
