490 Mr. A. M c Aulay on Quaternions as a 



Now 



Xi'-VVi'^-x'-'Xi'Vi' [•.•%/- , %iVi' = Vl = 0] ; 

 and, as will be proved directly, ™>\~ 1 XlV\"®) so ^ na ^ 



Hence 



Tins theorem is a particular case of one of several allied 

 quaternion theorems that I have found very useful in Physics, 

 but which have not yet been published. The proof just given 

 is not the simplest or most natural quaternion proof, but is 

 the simplest I can furnish when the theorem is divorced from 

 what may be called its natural surroundings. 



It still remains to prove that wi 1 -I ^ 1 , \7 1 , = 0. If it were 

 assumed, which is true, and which ought to be thoroughly 

 familiar to every mathematical physicist, that 



2m- 1 tfa)= — V/Oi/OsScov/Va'j 

 the statement would be obvious. As unhappily, however, this 

 is not generally known, the following indirect method of 

 proof may be adopted. Take p and p ! as the coordinate 

 vectors of any point in two different positions. Then 



=§TJvd8=$m- 1 x''Uv , ds! - jjj %" VVi'*' [equation (2)] . 



This being true for any space, is true for a single element c/?', 

 and therefore w 1 " 1 % 1 , v 1 ' = 0. The assumption that 



Uv^ = m-yUvW 



is easily seen to follow from the definition of m. For UW.s 

 may be taken as Ydp b dp c and JJv'ds' as VdpJdpJ, so that the 

 definition of m gives 



Sd Pa Uvds = m-^dpJXJv'ds' = m-^dpjJTWds'. 



IV. An Integral of the Equations of Fluid Motion. 



The integral I am about to give I do not propose to prove, 

 because what seems to me the best quaternion proof requires 

 properties of quaternions not proved in Tait's ' Quaternions.' 

 It has been said over and over again that quaternions in Physics 

 are only useful for expressing the results obtained by other 

 processes, and, perhaps, occasionally for furnishing a neater 

 proof of a truth when it has been discovered by other means. 

 To persons holding such views, of course there can be no 

 difficulty in furnishing a Cartesian proof of a quaternion 

 theorem. I will enter very fully into detail in the statement 

 of the- theorem, to leave no doubt as to the meaning. 



