Quaternions in Physics. 87 



in a science such as mathematics, which is purely an outcome 

 of logic ? 



In quaternions, a calculus uniquely adapted to Euclidian 

 space, this entire freedom from artifice and its inevitable 

 complications is the chief feature. The position of a point 

 (relative of course to some assumed origin) is denoted by a 

 single symbol, which fully characterizes it, and depends upon 

 length and direction alone, involving no reference whatever to 

 special coordinates^". Thus we use p (say) in place of the 

 Cartesian x, y, z, which are themselves dependent, for their 

 numerical values, upon the particular scaffolding which we 

 choose to erect as a (temporary) system of axes of reference. 

 The distance between two points is 



instead of the cumbrous Cartesian 



{(.r-y) 2 + (y-y) 2 + (~*-*') 2 P- 

 But the distance in question is fully symbolized as to direction 

 as well as length by the simple form 



p -p>. 



If three conterminous edges of a parallelepiped be p, p' ' , p ,r i 

 its volume is 



-S.pp'p". 



Even when advantage is taken of the remarkable conden- 

 sation secured by the intensely artificial notation for determi- 

 nants, Cartesian methods must content themselves with the 

 much more cumbrous expression 



X 



y 



z 



x' 



y' 



z' 



x" 



y" 



yll 



As we advance to higher matters, the Cartesian complexity 

 tells more and more ; while quaternions preserve their sim- 

 plicity. Thus any central surface of the second degree is 

 expressible by 



S/d0/d=-1, or T<^ = 1; 



while the Cartesian form develops into 



A.* 2 + 2B"#y + Ay + 2Wzx + 2Byz + M'z 2 = 1. 



The homogeneous strain which changes p into p' is expressible 



* Note here that though absolute position is an idea too absurd even 

 for the majority of metaphysicians, absolute direction is a perfectly defi- 

 nite physical idea. It is one essential part of the first law of motion. 



