of a Tangent-galvanometer. 139 



needle; and the absence from them of u, /3, 7 and h, teaches us 

 that in the ordinary form of tangent-galvanometer, a slight de- 

 rangement of the position of the needle in its cradle or in the 

 position of its point of suspension will not cause sensible error. 



2. In Gaugain's galvanometer a = |, and therefore from equa- 

 tion (5), 



z=tan^.^-^-<ri + |.(2« + /Stan6' + 3Ssin^)\ . (7) 



Hence in this description of galvanometer we get rid of the 

 trouble of applying a correction for the length of the needle ; 

 but it becomes necessary to attend carefully to the position of 

 the needle in its cradle, and to the horizontal adjustments of the 

 point of suspension, lest errors should creep in of which it would 

 be impossible to make any exact estimate. Accuracy of adjust- 

 ment in a vertical direction is of less importance, since 7 enters 

 only in the second and higher powers. 



To convey a clearer idea of the amount of error which may be 

 expected to arise in using Gaugain's galvanometer, let u, /3, 7, 

 and 8 be supposed each to have attained the magnitude 001 *. 

 Using this value in equation (7), we find that when the needle 

 deviates 20°, the error of the observation would rise to more 

 than jj)th. of the whole amount, at 32° to above -^'oth, and at 

 51° to above y^jth. No doubt these numbers are higher than 

 need be pi'ovided against by any careful manipulator ; but they 

 suffice to show distinctly that the errors arising in this way can- 

 not be safely disregarded, and that in conducting investigations 

 in which accuracy is a point of much importance, the ordinary 

 form of tangent-galvanometer is to be preferredf. 



* This is equivalent to assuming that a millimetre and a half is an 

 amount which each displacement could be su])posed to attain in a galvano- 

 meter whose current-circle has a diameter of three decimetres; or in Bri- 

 tish measure, displacements of about a sixteenth of an inch in galvanome- 

 ters a foot across. 



t In this investigation the magnetism has been conceived as collected 

 into two points, the poles of the needle. In general, however, the mag- 

 netic intensity is some function -^ of u, the distance from the centre of the 

 needle ; whence using 21 for the length of the needle, it is easy to see that 

 the only change we need make is to write in equations (4), (5), and {(')), 



\ u-^{u)du 

 " — . . instead of X-. This expression is obviously of the second 



order, and rejecting higlier orders, becomes 5 P if we assume the magnetic 

 distribution to be that of a linear magnet, in which i\r{u) is ap|)roximutely 

 equal to A^(e«"— e-«"). (Sec Biot's Physique, vol. iii. p. 77 .) 



