1867.] 



Currents hy Mechanical Work, 



401 



When the ratio of _^ to Hes between that of n to m and that of m to 

 r hes between +1 and —1, and the current must decay; but when pi is 

 less than g-m, a reciprocating current may be kept up, and will increase 

 most rapidly when 



q m\ V In J 



and 



— ('-r* 



When the commutator is of the second kind, the first step is to close both 

 circuits, so as to render the currents in them independent. The second 

 circuit is then broken, and the current in it is thus stopped. This produces 

 an elfect on the first circuit by induction determined by the equation 



L^+My=La;' + M?/' (14) 



In this case M=— M^, y=^x, and y'=0, so that 



(L~M)^=La?'; (15) 



where ^is the original, and-r' the new value of the current. 



The next step is to throw the circuits into one, M being now positive. 

 If x" be the current after this operation, 



(L + M)^'= (L + 2M + N)^" (16) 



The whole effect of this commutator is therefore to multiply the current 

 by the ratio 



L(L + 2M4-N)' 



The whole effect of the semirotation is to multiply the current by the ratio 



L+2M+N 

 L-2M+N*' 



The total effect of a semirevolution supposed instantaneous is to multiply 

 the current by the ratio 



^ L(L-2M+N)* 



Ifp and q be the number of windings in the first and second coils respec* 

 tively, the ratio r becomes 



l{lp~ — 2mpq + ncf) * 

 which is greater than 1, provided 2lmp is greater than {ln-{-m'^)q. When 



