100 PROCEEDINGS OF THE AMERICAN ACADEMY. 



ordinate {PQ) of the curve and the instantaneous value of the induced 



current by the distance {NQ) at that time, of the curve from the 



asymptote. The whole "amount" of the induced current up to 



the given time would be represented by the shaded area {A) shut 



in by the curve, the asymptote, and the ordinates, ^ = 0, t = OP. If 



the electromotive force were suddenly shunted out of the circuit 



after the current had reached its final value, the "extra current" 



would have the value 



E '■* 



-..-Z. (8) 



Helmholtz also studied the " forms " of the currents induced in the 

 secondary circuit of a small induction coil at the making and breaking 

 of the primary circuit, and, by using in the apparatus iron cores, some of 

 which were solid and some finely divided, he showed that the effect 

 of eddy currents in the iron upon the apparent duration of the induced 

 currents might be very appreciable. The results of Helmholtz's 

 experiments were confirmed with the aid of other apparatus, during 

 the next thirty years,^ by a number of physicists. 



The mathematical treatment of the subject begun by Neumann and 

 Helmholtz was in 1854 pushed somewhat farther by Koosen, and in 

 1862 E. du Bois-E,eymond "^ published an elaborate discussion of the 

 equations laid down by Helmholtz for the determination of the cur- 

 rents in two neighboring circuits of constant self-inductances {Li, L^ 

 and constant mutual inductance {M), and gave the solutions of the 

 simultaneous equations * 



(4) 



^■li + ^'-m + '■''■■" '^'' 



corresponding to a number of different sets of physical conditions, 

 in nearly the forms in which they now appear in textbooks. Du 



2 Felici, Ann. de Chimie, 34, 1852; N. Cimento, 3, 1856; 9, 1859; 12, 1874; 

 13, 1875. Cazin, Compt. Rend., 60, 1865 ; Ann. de Chimie, 17, 1869. Guillemin, 

 Compt. Rend., 50, 1800. Berlin, Mem. de la Soc. des Sc. Nat. Strasbourg, 6, 

 1865. Bazzi and Corbianchi, N. Cimento, 4, 1878. Bartolli, Mem. d. Ace. d. 

 Lincei, 6, 1882. Bazzi, Att. d. Ace. d. Lincei, 6, 1882. Lemstrom, Pogg. Ann., 

 147, 1872. V. Ettingshausen, Pogg. Ann., 159, 1876. 



3 Koosen, Pogg. Ann., 91, 1854. E. du Bois-Reymond, Monatsberichte d. Berl. 

 Akad., 1861, 1862. Brillouin, These, 1880; Jour, de Phys., 10, 1881; Compt- 

 Rend., 1882. 



