210 Prof. Ayrton and Dr. Sumpner on Alternate 



The formula employed with such methods for giving the 

 mean watts, whether it involves the reading of one instrument, 

 as in the case of the wattmeter (fig. 9), or of two instru- 

 ments, as with the methods illustrated in figures 4, 5, 6, 7, 

 and 8, or of three instruments, as with the methods illustrated 

 in figures 1, 2, and 3, gives with perfect accuracy r times the 



mean product of two currents, or - times the mean product 



of two P.D.s. Whether this mean product is directly pro- 

 portional to the mean watts given to ab depends in all the 

 nine cases on the following consideration : — 



The mean product between two currents which are sine 

 functions of the time is, as every student now knows, equal to 

 half the product of their maximum values into the cosine of the 

 phase angle between them. Therefore if the angle of lag be- 

 tween the current in ab and the P.D. between its terminals be 

 0, and the angle of lag between the current in cd and the 

 P.D. between its terminals be <£, and if the maximum values 

 of the currents in these two circuits be A/ and A/ respec- 

 tively, and the maximum values of the P.D.s at the terminals 

 of these circuits be V/ and Y/, it follows that the formula 

 used to measure the watts in the cases 2, 3, 5, 7, and 9 

 gives 



A/A/ cos (<9- (/)) 



and in the cases 1, 4, 6, and 8, 



V/V/cosCfl-fl) 

 2r 



But what we want to measure is the mean product of the 

 current in ab into the P.D. between its terminals, and this 

 product equals 



A/V/ cos 

 2 

 But 



rA 2 / =Y 2 / cos^); 



and in the methods illustrated in the figures 2, 3, 5, 7, and 9 



v/=v/, 



while in the methods illustrated in the figures 1, 4, 6, and 8 



A 3 =A 1 ; 



