(: 



a Geometrical Problem in Magnetism. 283 



equal and opposite to the same thing produced by^ at n, we 

 obtain 



m _ n .-jv 



(d 2 + m 2 )i~(d 2 +n 2 )i 



as the equation to find d, all the other measurements being- 

 known. 



Manipulating this expression, we have at length 



%mn) 4 2mn 4 2mn v ' 



Now it happens that in hyperbolic trigonometry we have 



cosh 3 6 - 1 cos 0-i cosh 3£ = (3) 



If, therefore, we make 



^±^ = cosh30, (4) 



2mn 



we have also 



^- = cosh d (5) 



By means, therefore, of the table of hyperbolic sines and 

 cosines which this Society has recently published, we can 

 easily determine d. 



We can deduce the distance of the poles apart, by applying 

 this proposition inversely. 



It is of course very easy to arrive at this state of things 

 practically with a magnet. 



Suppose we arrange a small magnetometer-needle in the 

 meridian, and notice the position of the light-beam on a scale 

 in an ordinary way. 



Find the direction of the axis of the needle by the condi- 

 tion that a long magnet laid in that line will not affect the 

 position of the spot of light ; then place that magnet, or any 

 other, at right angles to this position and move it in the 

 direction of its length until again the spot of light is at its 

 old place. No torsion affects these observations ; indeed, the 

 real meridian need not have been the direction found. 



Then, obviously, of the various quantities quoted in the 

 problem, we are in direct possession of d and m+n (the latter 



because the middle point of the magnet is distant — - — ( = Z), 



