466 Dr. W. E. Sumpner. The Vector Properties of 



deducing from the network figure the electrical power absorbed by 

 the conductors, as well as the magnitude of each current, and 

 voltage, that is chiefly due the usefulness and importance of this 

 mode of representation. 



It will be convenient here to describe briefly the mathematical 

 basis on which the method rests. It will be evident that we may 

 represent the value of a simple harmonic function Acosp£ either by 

 the projection of a revolving vector on a fixed line, or by the projec- 

 tion of a stationary vector on a revolving line. It will be more con- 

 venient for our purpose to adopt the latter plan. If, therefore, we 

 have a vector a, of length A, the projection of this on a line supposed 

 to revolve uniformly will always be equal to A cos, pt, provided the 

 -period of revolution T is such that pT = 27r, and provided also that 

 the line coincides with the vector at the instant from which t is 

 measured. We can, moreover, represent the value of B sinpt by the 

 projection at the same instant on the same revolving line of a vector 

 /3 of length B, provided only that the vectors a and ft are perpen- 

 dicular. It will also be evident that any other simple harmonic 

 function of the same period 



m A cos pt -f nB sin pt 

 will be similarly represented by the projection of the vector 



ma+n/3, 



and the length of this vector will be equal to the maximum value of 

 the function it represents. 



I\ow, in any system of conductors traversed by alternate currents, 

 if every electromotive force E^ is a simple harmonic function of one 

 period, E^ = Y x sin (pt + <fix), and if the coefficients of induction and 

 the capacities are all constant quantities, it follows mathematically 

 that each current, and each voltage, can be expressed in the general 

 form 



E = \ cos pt+[i sin^tf, 

 which can thus be represented by the vector 



v — + 



where a and /3 are the perpendicular vectors representing cosj?£ and 

 sinpt. 



If a and /3 are unit vectors, the length of v is ^/ (\ 2 + /* 2 ), which is 

 the maximum value of the function E. The magnitude of an alter- 

 nate current quantity is however estimated by the square root of its 

 mean square, and not by its maximum value, and since 



