Quaternions in Physics. 91 



Here the symbol ~d is called in, to effect a slight simplifica- 

 tion ; and we go a little further in the same direction by 

 putting u, v, w for 



dx dy dz 



dt'di'di' 



which are obviously the components of velocity of the point 

 for which Q is expressed. Thus we write 



|?=©-(g)-(f) + "0 



Of course you all know this quite well ; and you may ask 

 why I thus enlarge upon it. It is to show you how com- 

 pletely artificial and unnatural are our recognized modes of 

 expression. 



Fresnel well said : — La nature ne s^est pas embarrassde des 

 difficultes oVanalyse, elle na dvitd que la complication des 

 moyens. Why should we not attempt, at least, to imitate 

 nature by seeking simplicity ? 



The notation 8, as commonly used, is (like the d in dQ 

 above) quite unobjectionable. At least we cannot see how to 

 simplify it further. Its effect is to substitute, for any one 

 point of a figure or group, a proximate point in space, so that 

 the figure or group of points undergoes slight, and generally 

 continuous, but otherwise wholly arbitrary displacement and 

 distortion. It thus appears that d and 8 are entitled to take 

 their places in a calculus, such as quaternions, where sim- 

 plicity, naturalness, and direct intelligibility are the chief 

 qualities sought. We have now to inquire how such expres- 

 sions as 



jdQ, dQ dQ, 



l-j2 + m-j- + n ~ 



ax ay dz 



can be put in a form in which they will bear their meaning 

 on their face. 



It was for this purpose that Hamilton introduced his sym- 

 bol v. No doubt, it was originally defined in the cumbrous 

 and unnatural form 



. d . . d , , d 

 dx J dy dz 



But that was in the very infancy of the new calculus, before 

 its inventor had succeeded in completely removing from its 

 formulae the fragments of their Cartesian shell, which were 

 still persistently clinging about them. To be able to speak 

 freely about this remarkable operator, we must have a name 



