1899-1900.] Prof. Macfarlane on Hyperbolic Quaternions. 169 
Hyperbolic Quaternions. By Alexander Macfarlane, 
Lehigh University, South Bethlehem, Pennsylvania. (With 
a Plate.) 
(Read July 16, 1900.) 
It is well known that quaternions are intimately connected with 
spherical trigonometry, and in fact they reduce that subject to a 
branch of algebra. The question is suggested whether there is 
not a system of quaternions complementary to that of Hamilton, 
which is capable of expressing trigonometry on the surface of the 
equilateral hyperboloids. The rules of vector-analysts are approxi- 
mately complementary to those of quaternions. In this paper I 
propose to show how they can he made completely complemen- 
tary, and that, when so rectified, they yield the hyperbolic counter- 
part of the spherical quaternions. 
The celebrated rules discovered by Hamilton are : — 
i 2 --l j 2 =- 1 k 2 = - 1 
ij — k jk — i hi = j 
ji = — k kj = - i ik = —j. 
This is the statement of the rules as enunciated by Hamilton ■ it 
supposes an order of the symbols from right to left. When the 
order is changed to that from left to right, they become : — 
i 2 — - 1 
i 2 =- 1 
k 2 = 
-1 
ij=-k 
1 
II 
ki — 
ji — k 
¥- i 
ik — 
k. 
1 by vector-analysts are : — 
* 2 = +i 
i 2 = + i 
k 2 = 
+ 1 
ij = k 
jk= i 
ki = 
, f 
1 
Ji 
kj= - i 
ik — 
- j , 
and they suppose an order from left to right. They lead to pro- 
ducts in which the manner of associating the factors is essential, 
in this respect differing from the rules of quaternions. Can they 
he modified so that the order of the factors will be preserved, 
