532 Mr. A. 0. Allen on 



we have upon short-circuiting : — 



(L4-n)(^-/) + (P + R + W)^ + ^-.?/^ 



+ Wy -R,z—Mu + Uu=0, 



L(i-/) + (P + R)^-(Q + S)y-(R + S>+(R + S)M=0, 



M(i-/)-Ra?+M(y-|/) + Sy+.(R + S)xr 



-Ni-(R + S + T)u = 0. 



The determinant for the various periods is here a quartic in 

 D, so that the c's will not vanish unless the initial values of 

 z. z, z, and z are all zero. Of course x, y, z, and u are 

 initially zero ; 



• P 



finally, ^F = (R+S) u — Sy, which in general is not zero, so 

 that as a rule the balance is not' continuous. But if, in 

 addition to fulfilling the conditions for aggregate balance, 



L (P + 0)2 

 viz. PS = QR and ^ — /^m ? we can also satisfy the 



N P-f Q 

 condition ^ = — ^— by backing up the coil N with some 



extra self-inductance remote from the coil n, z then vanishes, 

 and the balance becomes a continuous one ; and, as must 

 invariably happen in such cases, the determinant factorizes. 

 Its factors are 



^(ND + T)(nD + W)~^-(MD-P-Q-R)+P(Q + S) 



and (^D + G)(ND + R + S + T+^+(ND + T)(R + S); 



the latter quadratic represents two exponentials of which 

 the coefficients are zero in a?, y } and u, so that these quantities 

 really consist of only two exponentials, corresponding to the 

 other quadratic. 



§ 14. To sum up this discussion, we may notice that in 

 five of the twelve methods considered (§§2, 5, 0, 7, 11) a 

 continuous balance is of necessity attained, if a balance is 

 attained at all; in four (§§ 3, 4, 8, 12) such a balance 



