EQUIVALENT SINE CURVES. 79 



if r is any integer whatever, and 



,.a t sin (rpt 0,) sin (spt B t )dt = 



M 



P I p 



2-J o 2 A si, 



if r and s are either both even or both odd. But as we have only 

 to consider the case in which r and s are always odd, the mean 

 square value of y is 



a* + fl 3 2 + as 2 -f . . etc. 

 2 



therefore the E.M.S. value of y is 



4- 3 2 + % 2 4- . . etc. 



which is independent of 0i, 2 , #3, etc. ; that is, the E.M.S. values 

 of functions containing only the fundamental and odd (or even, 

 but not both) harmonics depend solely on their maxima values, 

 and are independent of their relative phases. 



The E.M.S. value of any alternating current or E.M.F. can 

 therefore be represented by a definite vector. 



53. Root Mean Square Values of Alternating 1 

 Currents and E.M.F.s can be eompounded as 

 Vectors, whether they be Simple Sine Functions 

 or not. 



We now proceed to show that all E.M.S. values of alternating 

 currents and E.M.F.s can be compounded as vectors, whatever be 

 the shape of the curve representing them. 



In Fig. 30, let AB be a non-inductive resistance R, and BC a 

 coil having resistance and self- ^ R 



induction. A/vvvv nnnnnnp| 



Let vi, v%, and v be the corre- 



sponding instantaneous values 



5 , , -r, t -,0 



of the P.D. between A and B, 



B and C, A and C respectively, and let the respective E.M.S. 

 values be V\ t F" 2 , and V. 



Let i be the instantaneous current passing through the circuit, 

 and P the power given to the inductive circuit BC. 



Then 



V = Vi + #2 



therefore 



FIG. 30. 



