Mr. G. Boole's Notes on Quaternions. 279 



duced to a consistent theory ; for there undoubtedly exists a 

 theory of signs applicable as well to the signs of common dis- 

 course as to the signs of mathematics. Without attempting 

 to exhibit any complete doctrine on the subject, I will venture 

 to state one or two views which I have been led to form, and 

 apply them to the subject of quaternions. 



Signs employed as instruments of reasoning may, in one 

 point of view, be considered as the representatives of opera- 

 tions. This is not indeed the interpretation that we necessarily 

 attach to them, but it is one which it is probable that in all 

 cases we may attach, and from which their laws may in all 

 cases be deduced. If we employ a particular symbol or com- 

 bination of symbols A to represent a given operation, and 

 another symbol or combination of symbols B to represent 

 another given operation of the same kind, then, according to 

 the Arabic order of writing, AB will represent the successive 

 performance of the operations denoted by B and A. In order 

 that such signs may be really available as instruments of de- 

 duction, it is necessary that the laws of the symbols entering 

 into B and A should be such, that the expression AB deve- 

 loped according to those laws may, in agreement "with the con- 

 ventions established for the interpretation of A and B, repre- 

 sent an operation, the effect of which is equivalent to the com- 

 bined effects of the operation A and B performed in the order 

 (proceeding from right to left) AB. 



In strict accordance with this principle, we may assign an 

 interpretation to a quaternion to 4- ix +jy + kz, subject to the 

 condition 



ttJ 2 + a? 2 + y +2 2_ :1 ( 1# ) 



For if we write 



w=cos- #=sin — cos<p j/ = sin — cosy z=sm — cos^;, 



and assume the quaternion w + ix+jy + kz to represent a ro- 

 tation through an angle round an axis whose direction cosines 

 are <p, \J/, ;£, then representing this quaternion by A and any 

 other quaternion subject to a similar condition by A', we shall 

 have 



AA'=A", 



A" being a quaternion which, according to the same conven- 

 tions, will represent a single rotation equivalent in effect to the 

 two rotations represented by A' and A performed in the given 

 succession. 



A quaternion which does not satisfy the condition (1.) can- 

 not be directly interpreted in geometry. Such expressions 



