Professor Hermann finds the complete primitive of this differential equation, and then, introducing various values of r and the function E = e f(t), draws, by a process which is indeed laborious, the curves of the corresponding excursions. My own method, gives a good deal of the information so obtained in a much simpler manner.
Adopting the letters used by him, when / vanishes we have
dpjdt + rp = 0,
that is to say, whenever the E.M.F. falls to zero the reduced values of the subnormal and the radius vector are equal, but of opposite sign, and the curve, therefore, can never come back to the zero line under the action of a current which pulsates but does not alternate (see figs. 2 and 4 in Hermann’s paper). When the meniscus crosses the zero line, rp = 0, and dpfdt = re f(t), the impressed E.M.F. is then directly proportional to the subnormal. This involves the further fact that the crossing of the zero line by the meniscus must always lag behind the change of sign of the E.M.F.
If dpjdt vanishes, as it does at the apex of a spike or the bottom of r a notch, the instantaneous value of the impressed E.M.F. is directly proportional to the distance of the meniscus from zero.
The curves drawn by Professor Hermann are for the most part, so far as the eye can judge, similar to those obtainable under like conditions with the capillary electrometer. I have photographed and analysed many such, using rheotomes and dynamos of various kinds, both alternating and direct current, as sources of E.M.F. I have proposed, in a paper which has been in the publisher’s hands since last November, that this method should be used to determine the characteristic current curves of dynamos.*
All the confusing influence of the lag vanishes when such curves are analysed—there is no need to trouble about the equation to the ' curve, since each several term of its differential equation at any given point is found at once by my mode of analysis. But I must point out that an error has crept into Professor Hermann’s rendering of the curve given in fig. 6—or, rather, as it only pretends to be an approximation, that it is not equally accurate throughout. The portion c'd', which corresponds to a diminishing negative (below zero) potential is represented as rising with increasing velocity instead of falling more slowly, as it should do. Yet, when this negative potential ceases, the curve commences to fall from to e' along the logarithmic curve of discharge. This is impossible. When ef(t) is negative, the algebraic sum of dp/dt and rp must be negative also if the fundamental equation holds good. Probably the straight line cd has been placed too far to the right.
- * The Electrician,’ July 17, 1896, et se%.
