160 Prof. Silvanus P. Thompson on 



We shall then have 



E^ndN.10- 8 , 



E 2 = wC 2 N.10- 8 ;- 

 whence 



E 2 /B 1 =C 2 /C 1 =i, 



whatever the values of speed and field may be. Further, 



K 2 = e 2 + r 2 i 2 ; 

 whence 



e 2 = kei — r 2 i 2 — kr^ x . 



Now assume that the work wasted in the armature in mecha- 

 nical and magnetic friction, and in producing eddy-currents, 

 is negligibly small compared with the work done in driving 

 the generator part. This is, in fact, merely assuming that 

 the machine is properly designed and constructed ; for in 

 such machines, there being no driving-belt, the only forces 

 except gravity are centrally balanced, and mechanical friction 

 at the bearings is a minimum ; and further, if the iron core 

 is of proper quality and quantity, and properly laminated and 

 insulated, the losses due to hysteresis and eddy-currents are 

 very small*. We shall then have the work E^ done by the 

 primary current (in unit time) equal to the work E 2 2 2 done on 

 the secondary current. Consequently i 1 = ki 2 . Inserting this 

 value, we at once get 



e 2 = ke x — (r 2 + k 2 r^)i 2 . 



This shows that everything goes on in the secondary circuit 

 as though the induced electromotive force was transformed 

 from the difference of potentials at the terminals of the pri- 

 mary circuit in proportion to the respective numbers of 

 windings on the armature, and as though there were then 

 added to the real internal resistance of the secondary circuit 

 a resistance equal to that of the primary winding multiplied 

 by the square of the coefficient of transformation. If there 

 are equal weights of copper in the primary and secondary 

 windings, the actual resistances of the two windings will be 

 proportional to the squares of the numbers of turns : hence 

 the effect of multiplying r± by k 2 is to make it equal to r 2 ; in 

 other words, the added virtual resistance is equal to the real 

 internal resistance. 



* If F be the waste work done against mechanical and magnetic fric- 

 tion, the formula becomes 



e 2 = ke l — i 2 (r 2 + k 2 >\ 4- FA" 2 2 ) • 



