446 REPORT — 1882. 



the suspended coil should be compact, and should he placed in the position of 

 maximum effect. 



There is a further incidental advantaf^e in this arrangement, which it is the 

 principal object of the present note to point out. The expression for the attrac- 

 tion involves as factors the product of the numbers of turns, the square of the 

 current, and a function of the mean radii of the two coils, and of the distance 

 hetween their mean planes. Now, as may he seen from the fact that the square 

 of a current is already of the dimensions of a force, this function of three linear 

 quantities is itself of no dimensions. In determining its actual value we should 

 in general be subject to three errors ; but when the position is such that the func- 

 tion (for two given coils) is a maximum, the result is practically dependent only 

 upon the two mean radii, and being of no dimensions can involve them only in the 

 form of a ratio. In order then to calculate the result, all that it is necessary to 

 know with precision is the ratio of the mean radii of the two coils. This ratio 

 can be obtained electrically, with full precision, and without any linear measure- 

 ments. For, if the two coils considered as galvanometer coils are brought coaxially 

 into the same plane, the ratio of their constants can be found by the known method 

 of dividing a current between them in such a way that no effect is produced upon 

 a smaU magnet suspended at their common centre. The ratio of the resistances 

 in multiple arc gives the ratio of the currents, and this again (subject to small cor- 

 rections for the finite size of the sections), gives the ratio of the mean radii. 



It appears that in this way aU that is necessary for the absolute determination 

 of currents can be obtained without measurements of length, or of moments of 

 inertia, or even of absolute angles of deflection. In practice it will be desirable 

 to duplicate the fixed coil, placing the suspended coil midway between two similar 

 fixed ones, through which the current passes in opposite directions. A rough 

 approximation to the condition of things above described will be quite sufficient. 



2. On the Duration of Free Electric Currents in an Infinite Conducting 

 Cylinder. By Professor Lord Rayleigh, F.E.S. 



Taking the axis of the cylinder as that of s, we suppose that the currents are 

 functions of v'(-i'' + y^)> or r, only, and flow in the circles r = constant. 



From the equations given in jNIaxwell's ' Electricity,' vol. ii. §§ 691, 508, 607, 

 610, 611, we may deduce for a conductor of constant fj. 



(d^ fV- d\ dc 



w^dy--^dz^y=^^^^-dt 



with similar equations for h and a. 



In the present case the magnetic forces h and a vanish, and c is a function of r 

 onlv. Thus 



or if c varies as e-"'. 



(d'' \ (l\ . ^dc 



the solution of which, subject to the condition of finiteness at the centre, is 



c = ^J"(,(,/47r/xnC . ?•) = AJ^Qir). 

 To determine the admissible values of n, we have only to form the con- 

 dition which must be satisfied at the boundary of the cylinder r = R. It is evident 

 that the magnetic force must here be zero, so that the condition is 



Jo(^4n-/iMC". K)=Q. 

 The roots of the function are, 



2-404, 5-620, 8-654, 11-792, &c. 



For the principal mode of longest duration 



c = ^J"„(2-404»-/ie) 



