478 Royal Society, 



respectively, the ratio r becomes 



Z2 2 2 2 



p — m q 



9 ~ l(lp 2 — 2mpq + ?iq 2 ) ' 

 which is greater than 1, provided 2lmp is greater than (ln-\-m 2 )q. 

 When _______ 



P— l ( n j_™\j_ l I n 2 . n n Q m 2 



we have for the maximum value of r 3 



2~ 

 r=l + l 



v, 



n n m" . n m 



ml I m I 



In the experiment of Professor Wheatstone, in which the ends of 

 the primary coil were put in permanent connexion by a short wire, 

 the equations are more complicated, as we have three currents in- 

 stead of two to consider. The equations are 



n*+ g(^+%)=Sy+gj (m*+%)=q* + Ikz, (i7) 



^ + y + ^=0, (18) 



where Q, K, and z are the resistance, self-induction, and current 

 in the short wire. The resultant equations are of the second degree ; 

 but as they are only true when the magnetism of the cores is consi- 

 dered rigidly connected with the currents in the coils, an elaborate 

 discussion of them would be out of place in what professes to be 

 only a rough explanation of the theory of the experiments. 



Such a rough explanation appears to me to be as follows : — 



Without the shunt, the current in the secondary coil is always in 

 rigid connexion with that in the primary coil, except when the com- 

 mutator is changing. With the shunt, the two currents are in some 

 degree independent ; and the secondary coil, whose electric inertia is 

 small compared with that of the primary, can have its current reversed 

 and varied without being clogged by the sluggish primary coil. 



On the other hand, the primary coil loses that part of the total 

 current which passes through the shunt ; but we know that an iron 

 core, when highly magnetized, requires a great increase of current to 

 increase its magnetism, whereas its magnetism can be maintained at a 

 considerable value by a current much less powerful. In this way 

 the diminution in resistance and self-induction due to the shunt may 

 more than counterbalance the diminution of strength in the primary 

 magnet. 



Also, since the self-induction of the shunt is very small, all instan- 

 taneous currents will run through it rather than through the electro- 

 magnetic coils, and therefore it will receive more of the heating effect 

 of variable currents than a comparison of the resistances alone would 

 lead us to expect. 



