practical Instrument of Physical Research. 479 



discredit into which quaternions have fallen among physicists. 

 In his ' Electricity and Magnetism' (2nd edit. § 10) he 

 remarks: — "I am convinced that the introduction of the ideas, 

 as distinguished from the operations and methods, of Quaternions 

 will be of great use to us in all parts of our subject." Now 

 though very many study quaternions in a sort of dilettante 

 way, very few who have not read and been struck with this 

 passage do so before their mathematical ideas and methods 

 are nearly or completely crystallized. Workers naturally 

 find themselves, when still inexperienced in the use of qua- 

 ternions, incapable of clearly thinking through them and of 

 making them do the work of Cartesian Geometry, and they 

 conclude that quaternions do not provide suitable treatment 

 for what they have in hand. They then grow rather disgusted 

 with these vexatious quaternions, and consoling themselves 

 with the reflection that Maxwell, before penning the above 

 extract, had had more experience than themselves, decide that 

 the subject only requires a superficial study to be rendered of 

 as great utility as it is capable. 



The fact is that the subject requires a slight development 

 before being applicable to many important physical questions, 

 and these physicists do not continue their interest or enthusiasm 

 for the subject sufficiently long to enable them to furnish that 

 development. 



Quaternions differ in an important respect from other 

 branches of mathematics that are studied by mathematicians 

 after they have in the course of years of hard labour laid the 

 foundation of all their future work. In nearly all cases these 

 branches are very properly so called. They each grow out of 

 a definite spot of the main tree of mathematics, and derive 

 their sustenance from the sap of the trunk as a whole. But 

 not so with quaternions. To let these grow in the brain of a 

 mathematician, he must start from the seed as with the rest of 

 his mathematics regarded as a whole. He cannot graft them 

 on his already flourishing tree, for they will die there. They 

 are independent plants that require separate sowing and the 

 consequent careful tending. 



These are the explanations that can be given of the arrest 

 in the development of quaternions that followed on the death 

 of Hamilton. 



It is now w r ell to describe what I believe quaternions can do, 

 and what should not be demanded of them in the researches 

 of Physics. It is quite certain that the views about to be 

 enunciated will be voted, to say the least, extreme, and it will 

 not be possible to justify them in a short paper like the present. 

 Still it seems proper to give them in all their nakedness. 



2K2 



