of a Tangent-galvanometer. 137 



Substituting this value for 14*^ + ?/'^ in equation (1), and ex- 

 panding the denominator, we lind that the moment from the 

 current's action on the north pole 



=2^,^ \-^ . (l - S^»^+J^ . cos ^) 



x|l-f-;J^.(B-ycos*) + i^-i.{B-y'cos«'-!ic.|. 



In multiplying tiiis out, all odd powers of cos j> may be rejected, 



since! .cl(j> .cos.-"' ^ (j> = 0. After integrating the other terms 



Jo 

 and reducing, the expression for the moment becomes 



;>/..Xcos^7r.[-i-g+^-i^-{2B^ + y^-Gr^tan^ + /^) 



x(^A2-2b)|+&c.], 



into which X enters implicitly through the values (2) of A, B, x, y 

 and y'. Now by simply changing the sign of \ throughout and 

 then the sign of the entire expression, we obtain the moment 

 from the current's action on the south pole ; so that as the sum 

 of the moments from the current's action on the north and south 

 poles of the needle is equal to the sum of the moments from the 

 earth's action, we have at once, by putting M for the horizontal 

 intensity of the earth's magnetism, the equation* 



2/iM\sin6' = 4;^iXcos6'7r.[-^3 + ^ ;^ • [2W+y"-{xyi^xie+y''') 



(|a2-2b)} + &c.], (3) 



on the understanding that all terms containing odd powers of \ 

 are to be omitted in expanding the quantity under the bracket. 

 It only remains to invert the equation thus found in order to 

 have the value of i in terms of 6, which is the ultimate object 

 of our rcsearcli. 



In effecting these operations, if we regard X as a small quan- 

 tity of the first order, and X^ «, /3, 7 as small quantities of the 

 second order, and reject higher powers, we shall findf 



* From this equation tlie term — '— lias been left out, since it changes 



Hign with X. 



t If the second j)o\vers of the disphicements », fi, y had been retained 

 as well as of X, we shouhl have had under the bracket in equation (4) the 



