138 Mr. G. J. Stoney on the Adjustments of the Needle 

 . , . M.(l+«^r /, 3 l-4a^ 



+ |-lf^-(2« + ^tan^)|, (4) 



which becomes Bravais' formula by erasing the terms containing 

 « and /3. 



In this formula the error occasioned by a separation between 

 the ])oint of suspension and the point midway between the poles 

 of the needle is mixed up with other errors ; it may, however, be 

 distinguished from them by calling this interval of separation h, 

 and writing « + S sin ^ instead of «, and/3 + S cos 6 instead of /3; 

 a and ^ will thus become the displacements of the point of sus- 

 pension, and not, as heretofore, those of the magnetic centre of 

 the needle. By this substitution equation (4) becomes 



e=tan^. g-_-^.^l-5-^j-^,.X .(l-5sm 6) 



+ |.j-^2.(2a+/Stan^ + 38sin^)|, ... (5) 



in which the effect of a faulty position of the needle in its cradle 

 is separated from that arising from a defect in the adjustment of 

 its point of suspension. 



Two cases only need to be examined in detail : that in which 

 a = 0, which is the ordinary form of tangent-galvanometer, and 

 that in which fl = ^, which is Gaugain's arrangement*. 



1. In the first case, since « = 0, equation (5) becomes 



i=tan^.|^- |l-|x^(l-5sin^^)l> . . (6) 



=Ktan6'(l-f^X2sin2^^^ 



K being a constant so long as the same needle is used. From 

 either of these formulae a table may be once for all computed of 

 the corrections to be applied in consequence of the length of the 



additional terms 



+ \ l-0f- ^-"■'^ -^'- ^'^-"^ '^'' ^^ + i (iW • ^-""^^ *" ^^'' 



of which the first part vanishes in Gaiigain's arrangement of tlie galvano- 

 meter, and the second in the ordinary form. Neither these, however, nor 

 the other terms of the fourth order containing aK- and ^X", couhl become 

 sensible with any probable error of adjustment. 

 * Comptes Rcndus, vol. xxxvi. p. 191. 



which 



