336 PROCEEDINGS OP THE AMERICAN ACADEMY 



of soft iron are complicated by the fact that in addition to the pulse 

 which reaches a given point at a given time, necessarily later than the 

 time of its starting, there is another pulse, due to the direct action 

 of the primary and adjacent parts ; a pulse which at great distances 

 becomes, as before stated, indefinitely large in comparison with the 

 former. This will be evident on considering the nature of the two 

 curves which represent the action of each pulse. The direct pulse 

 decreases, of course, as the cube of the distance increases ; while the 

 logarithm of the pulse propagated through the bar, as determined by 

 actual experiment, becomes regularly less in proportion to the dis- 

 tance. For a distance of fifty inches on our half-inch bar, whose 

 coefficient of retained magnetism was 0.77 per linear inch, we have 

 the direct pulse, by calculation, some three and three-fourths times 

 greater than the indirect ; so that, no matter what be its difference 

 of phase, the effect cannot greatly exceed a retardation of 15°, and the 

 point of maximum retardation, which was more than 40°, must lie 

 somewhere on the bar. This was, as we have seen, the fact. It is 

 easy to show, moreover, that this point cannot be nearer the primary 

 than the point where the ratio of the indirect pulse to the direct 

 (which it greatly exceeds) is at its maximum ; and the latter is found 

 by calculation to be at about 11.6 inches. The actual position of the 

 point of maximum retardation of the phase lay always between these 

 two limits. 



As regards the phase of the primary itself, it is easily shown that 

 since the increase of its magnetism determines an electromotive force 

 opposed to that of the battery, which it cannot exceed, its rate of 

 magnetization is determined for short intervals of time by that electro- 

 motive force ; and that, in consequence, the magnetism keeps pace with 

 the time. 



In the differential equation 



put t = 0, and we have 



the constant of integration being zero; this value of jn being a special 

 solution of the general integral, 



