on Galvanometers. 65 



where the tangents are perpendicular to the axis of the coil, 

 OA gives the smallest radius that a coil must have in the 

 plane passing through OAM at right angles to the paper, 

 whether the magnet be at N and its half length be NP, or the 

 magnet be at N" and its half length be N'T'". 



For our purpose, however, we have not to consider the 

 problem of magnetic needles of different lengths at different 

 distances from a fixed plane, but the converse problem of a 

 magnetic needle of a fixed length in a fixed position, which is 

 at different distances from different planes, the critical radius 

 of convolution in each of which we wish to determine. 



If NP were the fixed half length of the magnet, then, as 

 already seen, A would be the critical radius of a convolution 

 at a distance NO from N, and the critical radius of a convo- 

 lution for another distance can be obtained by imagining the 

 figure 3 reduced in the proportion of NP to N'T'", in which 



NP 



case a critical radius ^ 1/; ;/ x OA will be obtained at a dis- 



NP 



tance - Nr//p/// ON" from N. Or, lastly, if we take OA as the 



half length and position of our magnetic needle, the critical 

 radius of convolution at every distance from will be ob- 

 tained by taking every such ordinate as N'T"' and multiply- 

 ing OA by the ratio of OA to N'T'", and the distance from 



OA 



for which this is the critical radius is equal to ^r //pw x ON". 



A curve giving the locus of such critical radii of convolution 

 is shown in fig. 4 for a needle of length 2, shown therefore in 

 the figure about twelve times its full size, and we learn from 

 it that at a distance from the centre of the coil equal, say, to 

 0*4 of the half length of the needle, the smallest convolution 

 should have a radius about 0*75 of the half length of the 

 needle, and that the wire must not be wound close to the 

 axis until the distance from the centre along the axis is about 

 0-72 of the half length of the needle. 



With an ordinary reflecting-galvanometer the needle re- 

 quires to have a free angular space for turning of about 15°, 

 represented by the space between the two lines LOL/. In 

 this region the radii of the smallest convolutions must be a 

 little greater than theory allows, otherwise the needle would 

 touch the coils, and generally a sufficiently near approxima- 

 tion can be made to the cavity which theoretically ought to be 

 left unwound by making it an oblate spheroid with a polar 

 axis about 0*72 of its equatorial diameter, the latter being of 

 course slightlv larger than the length of the needle. 



Phil. Mag"$. 5. Yol. 30. No. 182. July 1890. F 



