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



I believe that Physics would advance with both more rapid 

 and surer strides were quaternions introduced to serious study 

 to the almost complete exclusion of Cartesian Geometry, except 

 in an insignificant way, as a particular case of the former. 

 All the geometrical processes occurring in physical theories 

 and general physical problems are much simpler and more 

 graceful in their quaternion than in their Cartesian garb. To 

 illustrate the meaning here to be attached to " theory " and 

 " general problem/' take the case of elasticity. What is 

 meant by the general theory of elasticity is well enough 

 known. What I mean by a general problem is illustrated by 

 Bt. Venant's torsion problem for any cylinder. The same 

 problem for a cylinder of particular form would be called a 

 particular problem. For such particular problems we require 

 of course the theories specially constructed for the solution of 

 particular problems, such as Fourier's theories, complex vari- 

 ables, spherical and ellipsoidal harmonics, &c. It will thus 

 be seen that I do not propose to banish these theories, but 

 merely Cartesian Geometry. 



To establish these views it would be necessary to make 

 good the following two statements: — 



(1) Quaternions are already in such a state of development 

 as to justify the practically complete banishment of Cartesian 

 Geometry from physical questions of a general nature. 



(2) Quaternions will in Physics produce many new results 

 that cannot be obtained by the rival and older geometrical method 

 at all. 



To establish completely the first of these statements, it would 

 be necessary to go over the whole ground covered by general 

 physical questions. This would require a treatise of no small 

 dimensions. 



It is the second statement that must be considered of the 

 greater importance. Unfortunately this, too, cannot be jus- 

 tified here. It has already been conceded that the subject 

 requires a slight development, and this of course would neces- 

 sitate a rather lengthy introduction. It is only after this 

 development that I believe startling physical progress will be 

 made by help of quaternions. 



To bear out in part the assertions, however, a few examples 

 of the application of quaternions to a variety of physical 

 questions will be given. Some of the results below are of 

 interest in themselves. They have not been chosen mainly on 

 this account, however, but to illustrate as widely as is possible 

 in a short paper the variety of the questions in which the 

 subject may be expected to prove useful. 



The following are the examples chosen: — (1) A theorem in 



