2 ")0 Mr. Michael Donovan on Improvements in Galvanometers, ^x. 



Hence the compound needle is acted upon as a single needle, whose magnetic axis lies in 



the direction of the diagonal of the parallelogram ahove mentioned, and whose magnetic 



moment is 



fi = M cos iia + M' cos W- (2) 



Accordingly, the diagonal of the parallelogram already referred to will represent in mag- 

 nitude 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, wo- "o> te denoted by a, we have 



pP = J/2 + 2MM cos a + M'K (3) 



" In the case of the astatic needle, a = 180 - 8, S being a very small angle, and 

 cos a = - cos 8 = - 1 + -58-, q-p. whence 



fj^ = {M - M'f + MM'^^. (4) 



Accordingly, whem M - M' is not a very small quantity, the second term may be neglected 

 in comparison with the first, and fi = M- 31', nearly. On the other hand, when 31-31 = 0, 

 we have ju = 31S. 



" Returning to (1), and substituting for «'„ its value, uo + a, we have 



- sin a ,_. 



tan«o = -jj ; (0) 



-jr, + COS a 



by which the position of the needle with respect to the magnetic meridian, when at rest, 

 is determined. In the case of the astatic needle the preceding equation becomes 



tSinUo = j^—^,.Ssml. (6) 



From this we learn, 



" 1. That the tangent of the angle of deviation of the astatic needle from the magnetic 

 meridian varies, cmteris paribus, as the angle 8, contained by the magnetic axes of the two 

 component needles. 



" 2. That, however small that angle be, provided it be of finite magnitude, the tangent 

 of the deviation may be rendered as great as we please, and therefore the deviation be 

 made to approach to 90° as nearly as we please, by diminishing the difference of the moments 

 of the two needles." 



