180 Proceedings of Royal Society of Edinburgh. [sess. 
Consider now a general triangle on the hyberboloid of one sheet 
(fig. 9). 
Let the three axes he 
£=cosh 0’/3 + J - 1 sinh 0'a 
?7 = cosh 0 r 'y + J - 1 sinh O' ‘a 
£=cosh J - 1 sinh 6" ’a. 
Then £rj = cosh 0 cosh O' cos j3y - sinh 0 sinh 6' (1) 
- cosh 0 sinh 0 ,% /3 a - cosh O' sinh O' ay (2) 
+ V - 1 cosh 0 cosh O' sin /3y m a (3) 
In this case the length of the normal part of the Sinh may be 
greater than, equal to, or less than the length of the components 
along the axis of revolution. For we have to compare — 
cosh 2 0 sinh 2 0' + cosh 2 0' sinh 2 0 - 2 cosh 0 cosh O' sinh 0 sinh O' 
cos /3y with cosh 2 0 cosh 2 0' sin 2 j3y. Let sin /3y= 0, cos /3y= 
— 1 ; then the former term is the greater. Let cos j3y = 0, sin f3y 
= 1 ; then the former term is the less. And the terms may he 
equal. In the former case the axis of £rj has the form 
cosh <f>'e+ J— 1 sinh <£' a 
and the section is hyperbolic. In the latter case the axis of £rj 
has the form 
J - 1 {cosh 0 cosh O' sin /?y* a + J - 1 (cosh 0 sinh O'" /3a + cosh O' 
sinh O' ay)}. 
The axis inside the brackets denotes an axis of the equilateral 
hyperboloid of two sheets, and the section is elliptic. 
As before 
££=(£?) (vO 
therefore cosh ££ = cosh £y cosh r]£+ cosh {Sinh £r) Sinh rj£} 
and 
Sinh £r) = cosh ??£ Sinh £rj + cosh £rj Sinh rjt, + Sinh {Sinh £tj Sinh 
