Waste Space near the Needle. 



495 



Measurements &c. ' (ii. p. 375), and because of the weight 

 attaching to the names of the authors putting forth the propo- 

 sition, its emendation seems worth the short space necessary. 

 Briefly stated, it is true that the turns occupying a small 

 volume near the centre of a solid coil would have a reversed 

 effect upon a needle of finite length, but the volume is 

 roughly from one quarter to one eighth only of that indi- 

 cated by Ayrton, Mather, and Sumpner. It is approximately 

 a central sphere whose diameter is about one-half the length 

 of the needle, or the equivalent in volume of such a sphere. 

 Thus this consideration does become entirely insignificant in 

 practical construction, the major part of such a spherical 

 space being inevitably left empty. The diameter, one-half of 

 21, is but a very rough approximation, first arrived at from a 

 simple qualitative experiment. It would be an instructive 

 and not a difficult task for a student to map out the space 

 experimentally, and to locate the actual bounding-curves for 

 the surfaces of equal efficiency for which Thomson and Max- 

 well have given the equation relatively to the field at the 

 centre without discussing the effect of a needle not indefi- 

 nitely short. This work I shall endeavour to have performed. 



The error in the demonstration arose 

 from the fallacy of considering the de- 

 flective moment upon the needle as due 

 to the intensity and direction of the field 

 at the magnetic poles merely, regardless 

 of the fact that the field over the re- 

 mainder of the needle is not the same as at 

 the poles either in direction or strength. 

 Let the dots a 1 a 2 represent the sec- 

 tion of the wire of one small turn, the 

 diagram being a horizontal central sec- 

 tion. Let n s be the needle of length 

 21, the turn thus being one of a diameter 

 much less than 21. The deflecting 

 moment due to a current in the coil, 

 assuming a very thin uniformly magne- 

 tized prismatic needle, will of course be 



2 I m ./cos 6 . ds, 



m being the strength of pole of any thin 

 transverse section or shell of the needle, 

 ds the thickness of that section, / the 

 field-intensity at that point, and 6 the 

 field-direction angle with the axis of the 

 coil. For the indicated position and any 



2 M 2 



i: 



