172 



Consequently, if two lines be taken from any point of the 

 vertical axis, parallel to the magnetic axes of the two needles, 

 and proportional to their magnetic moments, M and M\ the 

 diagonal of the parallelogram constructed upon them must 

 lie in the magnetic meridian, when the compound needle is 

 at rest. 



Again, if we substitute u = Uq-^v, u'-Uo'+v, in the ge- 

 neral expression of the statical moment, it becomes, in virtue 

 of (I), 



X (M cosmq + M' cosMo') sin v. 

 Hence the compound needle is acted upon as a single needle, 

 whose magnetic axis lies in the direction of the diagonal of 

 the parallelogram above mentioned, and whose magnetic mo- 

 ment is 



11= Mq.o%Uq^M' cos Mo'. (2) 



Accordingly, the diagonal of the parallelogram already re- 

 ferred to will represent in magnitude the magnetic moment 

 of the compound needle. For, if the equations (1) and (2) be 

 squared, and added together,, and the angle contained by 

 the magnetic axes of the two needles, Uq - m^, be denoted 

 by a, we have 



fx^ = M'^ + 2MM' cos a + M^ (3) 



In the case of the astatic needle, a = 180 -8, S being a 



very small angle, and cos a = - cos S = - 1 + ^ S^, q.p. whence 



,j.'' = {M-M'y + M M' g2. (4) 



Accordingly, when M - M' is not a very small quantity, the 



second term may be neglected in comparison with the first, and 



fi = M-M\ nearly. On the other hand, when M-M' = 0, 



we have ij.= Md. 



Returning to (1), and substituting for Uq its value uq + o, 



we hiave 



-sina ,_. 



tmuo=-^ ; (5) 



-j^, + cos a 

 M 



