332 ON THE THEORY OF COMPOUND MAGNETIC NEEDLE. 



whose magnetic axis lies in the direction of the diagonal of the 

 parallelogram above mentioned, and whose magnetic moment is 



fj. = M cos MO + M' GOSU'O. (2) 



Accordingly, the diagonal of the parallelogram already referred 

 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, u' - u , be denoted by a, we have 



t? = M 2 + 2MM' cos a + M '\ (3) 



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

 small angle, and cos a = - cos 8 = - 1 + \ S 2 , q .p. whence 



V = (H-M f y + MM'&. (4) 



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

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

 /u = H-M', nearly. On the other hand, when M - M' = 0, we 

 have n = MS. 



Eeturning to (1), and substituting for u' its value U Q + a, we 

 have 



to*- g-*" ; (5) 



- + COSa 



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 



tan w - -^fySsinl'. (6) 



From this we learn, 



1. That the tangent of the angle of deviation of the astatic 

 needle from the magnetic meridian varies, cceteris paribus, as the 

 angle, S, 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. 



