of the Conditions of Maximum Efficiency . 31 



induction. The phase-interval is then the only independent 

 variable in the problem, and what its value must be to give 

 the maximum efficiency of transmission of Power, and what 

 that efficiency will then be, are the questions to which we 

 shall have answers from Geometry alone. A short statement 

 of the general method of representation will make the 

 particular steps required for these problems perfectly clear. 



2. Let a straight line of fixed length, and situated in the 

 plane of the paper, undergo uniform rotation in that plane. 

 Then its projection upon a fixed indefinitely long line also in 

 that plane will undergo harmonic variation, and may repre- 

 sent any magnitude capable of undergoing such change (e. g. 

 an electromotive force), the maximum value of this varying 

 magnitude being represented by the revolving line itself. 

 The period in which the revolving line makes one complete 

 revolution is the period of the change. Hence, if we know 

 the position of the fixed line and of the revolving line at any 

 instant, we can say in what particular phase the magnitude 

 undergoing harmonic change is at that instant. For instance, 

 suppose these lines make 30° with each other, we can say at 

 once that the magnitude is removed from its maximum value 

 by an interval of time equal to one twelfth of the period. If 

 the angle is at the instant increasing, the magnitude has 

 passed its maximum value that interval of time ago. If the 

 angle is growing less, the magnitude will attain its maximum 

 after that interval of time. It is therefore necessary to fix a 

 positive direction of rotation as representing the positive lapse 

 of time. [That direction which is opposite to that of the 

 hands of a watch will here be adopted.] 



3. It follows that when we have two such electromotive 

 forces acting in the same circuit, having different maximum 

 values but the same Period, since each is represented by the 

 projection of a revolving straight line upon a fixed straight 

 line, the resultant electromotive force at the instant is the 

 algebraical sum of the individual projections. And if the two 

 revolving lines are laid down as the two sides of a triangle 

 taken in order, the rotation being uniform and the same for 

 both lines, the lines will remain always inclined at the same 

 angle to each other, and the algebraical sum of their projec- 

 tions is the projection of the third side. Thus, in the matter 

 of such electromotive forces, we have a theorem exactly cor- 

 responding to the triangle of directed quantities. 



4. We may extend this mode of representing such quantities 

 so as to form a theorem corresponding to the polygon of 

 directed quantities, and cite it thus : — 



