144 TRANSACTIONS OP SECTION A. 



But continual thought and envisaging of the subject is still required to give 

 anything approaching to instinctive appreciation such as we have in ordinary 

 Newtonian dynamics. I venture to say that the subject is pre-eminently one 

 for physicists and physical mathematicians. In some ways the new ideas bring 

 us back to Newton's standpoint as regards so-called absolute rotation, a subject 

 on which I have never thought that discussions of the foundations of dynamics 

 had said absolutely the last word. Some relativists would abolish the ether, 

 1 hope they will not be successful. I am convinced that the whole subject requires 

 much more consideration from the physical point of view than it has yet received 

 from relativists. 



The better the student of physics is grounded in the older dynamics, and 

 especially in the dynamics of rotation, the sooner will he be able to place himself 

 at the new point of view, and the sooner will his way of looking at things begin 

 to become instructive. 



With regard to the study of physics in our Universities and Colleges, I had 

 written a good deal. I have fut that aside for the present, and will content 

 myself with only a few general observations. First, then, it would, I think, be 

 conducive to progress if it were more generally recognised that dynamics is a 

 physical subject, and only secondarily a mathematical one. Its study should 

 be carried on in the departments of physics, not in those of mathematics or in 

 separate departments of applied mathematics. It is, or ought to be, essentially 

 a subject of the physical lecture-room and the physical laboratory, it belongs 

 in short to natural philosophy , but not to physics divorced from mathematics, 

 nor to the arid region of so-called applied mathematics, where nothing experi- 

 mental ever interrupts the flow of "blackboard analysis. The student should be 

 able to handle rotating bodies, to observe and test the laws of precession and 

 nutation, to work himself, in a word, into an instinctive appreciation of at least 

 the simpler results of rotational theory. He should learn to tliink in vectors, 

 without necessarily referring either to Hamilton or to Grassmann. Some people 

 appear to censure the use of vector ideas without the introduction at the same 

 time of some form of vector notation. I cannot agree with them. Personally 1 do 

 not feel drawn to any system of vectors in particular — all have their good points, 

 and in some ways for three dimensional work the quaternion analj'sis is very 

 attractive — but vector ideas are of the very utmost importance. 



Hence I deprecate the teaching, however elementary, which as a beginning 

 contents itself with rectilineal motion. The true meaning of rate of change of a 

 directed quantity, even of velocity and acceleration, is missed, and instead of 

 having laid a foundation for further progress the teacher, when he desires to 

 go beyond the mere elements, has not merely to relay his foundations, he has in 

 fact to extract imperfect ideas from hie pupils' minds and substitute new ones, 

 with the result that a great deal of avoidable perplexity and vexation is pro- 

 duced. The consideration of the manner of growth of vectors — the resultant vector 

 or it may. be component vectors, according to convenience — is the whole affair. 

 As a simple illustration of what I mean, take this : A vector quantity has a 

 Certain direction, and also a magnitude L. It is turning im a certain plane with 

 angular speed te. This turning causes a rate of production of the vector quantity 

 about a line in that plane and perpendicular to the former, and towards which 

 the former is turning, of amount Lw. Thus a particle moving in a curve with 

 speed v has momentum mv forwards along the tangent at the position of the 

 particle. The vector is turning towards the principal radius (length R) of 

 curvature at the point at rate v/R. Hence towards the centre of curvature 

 momentum is growing up at time rate mv-/R. 



Dealt with in this way, with angular momentum instead of simple momentum, 

 the motions of the principal axes of a rigid body give the equations of Euler 

 instantly and intuitively, and all the mind-stupefying notions of centrifugal 

 couples, end the like, are swept away. 



Witli regard to mathematics, the more the physicist knows the better, and 

 he should continually add to his store by making each physical subject he takes 

 up a starting-point for further acquisition. Some very philistiiie notions as to 

 mathematics prevail, and are very mischievous. For example, I once heard an 

 eminent practical engineer declare that all the calculus an engineering student 

 requires could be learned in an hour or two. This is simply not true, nor is 

 it (rue, as some exponents of ultrasimplicity seem to suggest, that the profes- 



