( 402 ) 



In these formulae c denotes tlie velocity of light in the aether. 



In the greater part of what folloAVs, we shall confine onrselves to 

 cases, in which the components of the above vectors and of others 

 we shall have occasion to consider, are harmonic fnnctions of the 

 time with the frequency n. Then, the mathematical calculations can 

 be much simplified if, instead of the real values of these components, 

 we introduce certain complex quantities, all of which contain the 

 time in the factor e"*^ and whose real ])arts are the values of the 

 components with which we are concerned. If 21 ;i, '^ly, ^\~ are com- 

 plex quantities of this kind, relating in one way or another to the three 

 axes of coordinates and in which the quantity e"'*' may be multiplied 

 by complex quantities, the combination {^\x, ^^y, '^1-) may be called a 

 complex vector ^\ and ^\x, '31^, '^l- its components. 



By the real part of such a vector w^e shall understand a vector 

 whose components are the real parts of -1^, ^\y, '^1-. It will lead to 

 no confusion, if the same symbol is used alternately to denote a 

 complex vector and its real part. It will also be found convenient to 

 speak of the rotation and the divergence of a complex vector, and 

 of the scalar product (^1, ^) and the vector product [31. 55] of two 

 complex vectors 51 and 35, all these quantities being detined in the 

 same way as the corresponding ones in the case of real vectors. 

 E. g., we shall mean by the scalar product (^. 5?) the expression 



It is easily seen that, if ^, ^, S and 93 are complex vectors, 

 satisfying the equations (1) and (2), their real parts will do so 

 likewise. The denominations electric force, etc. will be applied to 

 these complex vectors as well as to the real ones. 



One advantage that is gained by the use of complex quantities 

 lies in the fact that now, owing to the factor e'"', a differentiation 

 with respect to the time amounts to the same thing as a multiplication 

 by in; in virtue of this the relation between ^" and S and that 

 between S^ and 5) may be expressed in a simple form. Indeed, we 

 may safely assume that, whatever be the peculiar properties of a 

 ponderable body, the components of ^ are connected to those.of ^" by 

 three linear equations with constant coeflicients, containing the com- 

 ponents and their diiferential coeflicients with respect to the time. 

 In the case of the complex vectors, these equations may be written 

 as linear relations between the components themselves; in other 

 terms, one complex vector becomes a linear vector function of the 

 other. A relation of this kind between two vectors 51 and Q) can 

 always l)e expressed by three equations of the form 



