Proof of the Generality of certain Formulae* 355 



stants had to be employed in each case to reduce the arbitrary 

 scale-readings to absolute measure. We will, on the contrary, 

 suppose the instruments to be graduated as they ought to be, 

 so that if D , D s , and D be the readings of the three instru- 

 ments, they are equal respectively to the square root of the 

 mean square of the primary current, the square root of the 

 mean square of the secondary current, and the mean product 

 of the two currents. 



With this definition Mr. Blakesley proved geometrically 

 that the watts given to the primary coil of the transformer 

 were equal to 



P 



p d; + ^sJ) 



ps J 



where p and s are the resistances, in ohms, of the primary coil 

 and the whole secondary circuit respectively, and P and 8 

 are the numbers of windings in the primary and secondary 

 coils of the transformer. 



The formula is a simple one, and the values of the ex- 

 pressions in it are fairly easy to obtain experimentally. The 

 proof of the formula, however, as given by Mr. Blakesley, was 

 based on the following assumptions : — 



1. The variations of the primary and secondary currents 

 are harmonic. 



2. The variation of the magnetism of the core is harmonic. 



3. The magnetic stresses produced in the iron core by the 

 currents in the primary and secondary coils are directly pro- 

 portional to the ampere-turns in these two coils. 



4. Each turn in each coil embraces at any moment the same 

 number of lines of force. 



5. The secondary circuit outside the transformer is non- 

 inductive. 



II. 



In the spring of last year, 1890, Mr. Wightman, one of the 

 third-year students of the Central Institution, showed that an 

 analytical method for measuring the efficiency of a transformer, 

 which had been described in one of the lectures at the college, 

 could by a slight transformation be employed to prove the 

 generality of Mr. Blakesley's formula given above. 



The proof is quite simple, and shows that the formula in 

 question is true whatever function the currents or the mag- 

 netism be of the time, and whatever amount of hysteresis or 

 magnetic lag may exist. In fact the proof is independent of 

 Mr. Blakesley's assumptions Nos. 1, 2, and 3, mentioned 

 above. 



