Quaternions in Physics. 89 



which . occurs in connexion with the strain produced by a 

 given displacement of every point of a medium, is 



Say. S#Vi . "VWi. 

 Its Cartesian expression is, with the necessary specification 



made up of three similar terms of which it is sufficient to 

 write one only, viz., 



(aV — ba')(— — — — —\ + (ca'--ac')( — — — — d —\ 

 \dx dy dy dx) ^ \dz dx dx dz) 



Now, suppose this to be given as the ^-coordinate of a point, 

 similar expressions (formed by cyclical permutation) being 

 written for the y and z coordinates. How long would it take 

 you to interpret its meaning? 



Look again at the quaternion form, and you see at a glance 

 that it may be written 



Y(Sav.<0(S£V*), 

 in which its physical meaning is more obvious than any mere 



form of words could make it. 



Or you may at once transform it to 



-iS.(V«0)vVi'-V«»i, 



which shows clearly why it vanishes when a and /3 are 

 parallel. 



1 need not give more complex examples : — because, though 

 their quaternion form may be simple enough (containing, 

 say, 8 or 10 symbols altogether), even this unusually large 

 blackboard would not suffice to exhibit more than a fraction 

 of the equivalent Cartesian form. 



Any mathematical method, which is to be applied to 

 physical problems, must be capable of expressing not only 

 space-relations but also the grand characteristics (so far as 

 we yet know them) of the materials of the physical world. 

 I have just briefly shown how exactly and uniquely quater- 

 nions are adapted to Euclidian space ; we must next enquire 

 how they meet the other requirements. 



The grand characteristics of the physical world are : — 

 Conservation of Matter, with absolute preservation of its 

 identity ; and Conservation of Energy, in spite of perpetual 

 change of a character such as entirely to prevent the recogni- 

 tion of identity. The first of these is very simple, and needs 

 no preliminary remarks. But the methods of symbolizing 



