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V. On the Calculation of the Induction- Coefficients of Coils, 

 and the Construction of Standards of Inductance, and on 

 Absolute Electrodynamometers. By Andrew Gray, M.A., 

 Profesor of Physics in the University College of North 

 Wales*. 



IT is well known that the electrokinetic energy of the cur- 

 rents in two circular conductors can be expressed by a 

 series of zonal harmonics. This series, when used in the 

 ordinary way to find the energy of the currents in two cylin- 

 drical coils (and hence also the induction-coefficients of the 

 coils) by expansion of each term of the series and subsequent 

 integration, does not yield expressions which are convenient 

 for practical applications, as the work of calculation is long 

 and tedious. I have found, however, that it is possible very 

 simply to integrate each term without expansion ; and the 

 result shows that a pair of coils may be constructed in such a 

 way that the zonal harmonic expression reduces to a very 

 manageable form, and the energy of the currents, and there- 

 fore the induction-coefficients and mutual action of the coils, 

 can be very readily obtained. 



With regard to the construction of coils it is possible, by 

 using fine wire wound by a screw-cutting lathe in a close 

 single layer on an accurately turned cylindric surface, to 

 make a coil of a large number of turns the dimensions of 

 which can be determined very exactly, and in which the dis- 

 tribution of the wire is perfectly definite. Such a single- 

 laver coil I have long advocated for use in absolute gal- 

 vanometers. It has sufficient uniformity of field to render 

 the placing of the needle at the exact centre quite un- 

 essential, and it can be made sufficiently sensitive, so that it 

 possesses the advantages of the Helmholtz double-coil arrange- 

 ment, without the uncertainty which exists in the latter as to 

 the distribution of the different turns of wire in the two 

 multiple-layer bobbins, or requiring the correction-terms 

 which these bobbins involve on account of their finite cross 

 section. 



By integrating the zonal harmonic expression for two 

 circles with intersecting axes, in order to find the correspond- 

 ing expression for the mutual energy of two single-layer coils, 

 I have obtained a series, the even terms of which all vanish 

 when one at least of the coils is placed with its centre at the 

 intersection of the axes. The third term vanishes if the 

 smaller of the two coils is so placed, and has its length and 



* Communicated by the Author. 



