298 



suit is increased. In order to show this, the author en- 

 tered into an examination of the amount of the probable 

 error in the two methods, from which it appeared that the 

 probable error of Q, arising from an error in the observed 

 deflection, will be less than in the usual method in the ratio 

 of 1 to 5.563, even when the latter is employed in the man- 

 ner most conducive to accuracy. In fact, the ratio of the 

 probable error to the entire quantity is found to be ex- 

 pressed, in the two cases, by the formulae 



Aq _ Am Aq _ 1/9^" -4- I Am 



Q ~ U ' Q, ~ (f — 1 U 



where q denotes the ratio of the two distances; and \\\Qleast 

 value of the factor LJ—JL- is 5.563, and corresponds to the 

 ratio q = 1.32. 



In order to know the ratio h, it is necessary to de- 

 termine the moment of the force exerted by the deflecting 

 magnet upon the suspended magnet, extending the approxi- 

 mation to the terms involving the fifth power of the distance. 

 The axis of the deflecting magnet being supposed to lie in 

 the right line joining the centres of the two magnets, and 

 the axis of the suspended magnet forming the angle ^ with 

 that line, this moment is found to be 



2mm' . , S, . f r.^Z cr- 9 1 InM'sNI > 



— rsin^ Jl+(2-2-l-3(5cos2^-l)-;j-r,? ; 



D v.\M M/Dj 



in which m and m', M3 and M'3, denote certain integrals de- 

 pending on the distribution of free magnetism, in the deflect- 

 ing and suspended magnets, whose values are 



=r \ mrdr, M3 = \ mr^dr, 



m being the quantity of free magnetism in any transverse 

 section of the magnet, r its distance from the centre, and I 

 half the length of the bar. The form of this result exhibits 



