54 REPORT— 1908. 



If now dr denotes an element of a surface, as a vector normal to the 

 surface, theia 



7f d T =■ {7} d t) 

 gives the number of Ty-lines crossing the element and 



/r,d. 



extended over a finite area, the number which cross this area. 



The differentiation of a vector with regard to a scalar, say, the time, 

 is very simple and oflers nothing new, although some results are striking. 

 The variation of a function of p due to a displacement of the point or a 

 change of p requires Hamilton's operator ^. 



This operator is of the nature of a vector and can operate on a scalar 

 or a vector function, and on the latter in two ways. The three results 

 thus obtained are of such physical importance that Maxwell has given 

 them special names. From a scalar u we get the vector ^ m, which is 

 a vector existing in general at every point where u exists and is normal 

 to the It-surface through the point considered. Maxwell calls it the 

 slope of u. 



If the operand is a vector jj the two results are, in Hamilton's notation 

 with Maxwell's name, 



S \- »? = convergence of »/, 

 V V; = curl 7;. 



Instead of the former we have in vector analysis (^ 7/) = ^>; = - S y »;, 

 wliich has been called by Cliflford the divergence of »;, is written by Heavi- 

 side Div. ?;. I have found it convenient to introduce for Vv or ' curl ' a 

 special symbol, a v with an arrow-head rising from the top. 



A v-calculus has been worked out in connection with quaternions by 

 Tait, and recently by Professor Joly. The same can be done in vector 

 analysis, and a good deal has been done (by Heaviside, Gibbs, and others). 

 It deserves to be established as a purely mathematical theory. 



Various applications, partly physical, partly relating to pure mathe- 

 matics, Avere given at the meeting which are here omitted. 



A few words about the teaching of vectors at school. My idea is 

 that they should be introduced before trigonometry is begun, soon after, 

 ;ind in connection with the use of squared paper, by plotting points from 

 given position -vectors, and curves from simple vector-equations. 



The decomposition of a position-vector gives the co-ordinates of a 

 point together with their sense, and then the equations 



x^r cos 6, 2/='>' sin 9 



lead to general definitions of the trigonometrical functions holding for all 

 four quadrants. 



In the discussion which followed, and in which the President of the 

 Section, Professor Bolzmann, Professor Larmor, Sir Oliver Lodge, Dr. 

 Sumpner and others, and Professor Joly and Mr. Swinburne by letter, 

 took part, no voice was raised against the extended use of vectors, but 

 nearly everyone expressed the wish that an agreement should be come 



i 



