ON THE USE OF VECTORIAL METHODS IK PHYSICS. 63 



With regard to vectors as entering into the study of the relations 

 between physical quantities, Maxwell speaks against quaternionic opera- 

 tions, but he has no word against vectors. He never makes use of 

 quaternions in his great work, but in the second volume constantly uses 

 vectors, and gives at the end his final results in the form of vector 

 equations.' In the passage quoted he states clearly why he uses Cartesian 

 methods, and I cannot help thinking that he would have used vector 

 methods throughout if he had found ready to hand a vector analysis 

 instead of a theory of quaternions, and if such analysis had been common 

 property. 



At present every electrician expresses himself in terms of Faraday's 

 lines of force ; all elementary text-books use them, and by their aid the 

 elements of electricity and magnetism have been made extremely simple. 

 Theorems which formerly could be proved only by the aid of a consider- 

 able amount of analytical work are now proved in a few lines of reason- 

 ing, and often in a much more convincing manner. But when a certain 

 point is reached there is an hiatus. The more advanced parts of the science 

 are still only accessible by aid of the old methods of the differential or 

 integral calculus, using co-ordinates of points and components of forces. 

 These results are therefore inaccessible to all who have not been able to 

 spend years on pure mathematics. Most physicists and electricians have 

 neither inclination nor time to do this. To bridge over the hiatus and to 

 introduce continuity in treatment requires vector analysis. 



The subject itself is not difficult, and would become very easy if the 

 first elements of vector algebra (which are very simple) were introduced 

 into the school curriculum. 



Vector addition is already known from the composition of forces. There 

 come next two products of two vectors each — the scalar-product and the 

 vector-product. The former is simple enough, as it follows all laws of 

 common algebra with the exception of one which, although the law which 

 distinguishes it from all other algebi-as, is generally not even mentioned 

 in English text-books. It is the law that a product can vanish only if 

 one factor vanishes. The second, the vector- product, requires more care 

 in manipulation in so far as the commutative law does not hold. In addi- 

 tion to these, two products of three factors have to be considered, and the 

 whole algebra is complete. 



We have next to consider variable quantities. If m is a scalar-function 

 of the position of a point, hence of the position-vector p, then u = const, 

 represents a surface which may be called a w-surface. If u is one-valued, 

 through every point one such surface can in general be drawn. Thus 

 space becomes filled with these surfaces, which are constantly used undei 

 the name of equipotential, isothermal, <kc., surfaces. Similarly if 77 

 denotes a vector, varying from point to point, lines which may be called 

 7] lines are formed by drawing at any point the vector ?;, going along it 

 through an infinitesimal distance, and drawing here the new vector. 

 These are Faraday's lines of force in their purely geometrical aspect. 

 They give the direction of 7; at every point, but not the magnitude. 

 This Faraday introduced also in the well-known manner by drawing only 

 some of these lines so that the number of the lines which cross a given 

 area represents the magnitude of r/. 



See also his little book Matter and Motion, 



