GAUGAIN'S COIL. 105 



If the needle, which we assume is symmetrical, cannot be reduced 

 to two poles, it is readily seen that we should replace the length / 



2u/ 2u/ 3 



by -^ , and / 2 by -^r , /* being the magnetic mass at a distance 



2jfJ* 2sp,l 



A from the axis of rotation. 



The principal term in the expression for D, represents the 

 moment of the action which the needle would experience in a 

 uniform field, the strength of which was the same as at the point P. 



The first term of the correction within the bracket, is proportional 

 to the factors (1-5 sin 2 a.) This factor is positive and equal to unity, 



for a = ; it vanishes for sin 2 a = - , or tan a = - ; that is to say, for 



an angle of 26 4'. It then again becomes negative and attains the 

 value i '5, for a = 45. 



The second term proportional to (i - 14 sin 2 a + 21 sin 4 a) is zero, 

 for a = 4 40' and a = 3549'. This factor is positive and equal to 



unity for a = 0, passes through a minimum equal to - -, near 19 28, 



and resumes the same positive value near 4346'. In most cases this 

 term can be neglected. 



748. GAUGAIN'S COIL. Various devices may be made use of 

 to get rid of the first tertn of correction of the couple D, so that the 

 needle may be sensibly under the same conditions as if the field were 

 uniform. 



We might in the first case arrange the experiments in such a 

 manner that the needle always made an angle of about 26 with the 

 plane of the frame, which would make the factor (1-5 sin 2 a) 

 virtually zero. 



Another plan consists in placing the centre of the needle at the 



ci 



distance x , a condition which would annul the factor a 2 - ^x 2 . 

 2 



Gaugain had found experimentally* that a needle whose centre is 

 on the axis of a circular current, and at a distance from the centre 

 equal to half the radius, experiences deflections the tangents of which 

 are very exactly proportional to the strength of the current that is to 

 say, that the couple of the current is proportional to the cosine of the 

 deflection. Starting from this observation, he had coils made in the 

 form of the frustrum of a cone such that the tangent of the same 

 angle at the summit was equal to 2. If the centre of the needle be 

 placed at the apex of the cone the preceding condition is satisfied 



* GAUGAIN. Comptes rcndus de VAcad. de Science, Vol. xxxvi., p. 191. 1853. 



