208 Dr. A. C. Crehore on the Theory of the 



sufficient electrical conductivity to admit of carrying a cur- 

 rent, immersed throughout almost its entire length in a 

 powerful magnetic field, and we have the essential features 

 of the string galvanometer. The direction of the field is, of 

 course, approximately perpendicular to that of the string, 

 and the transverse motion of each element of the string due 

 to the mutual action between the current in it and the 

 magnetic field is in a third direction approximately perpen- 

 dicular to each of the other two directions. 



In this paper the string will be treated as if it were 

 immersed in a magnetic field the lines of which are parallel, 

 but the force may vary in intensity from point to point along 

 the string. As particular cases two different field dis- 

 tributions are considered, first, the uniform field which most 

 nearly represents the galvanometer as at present constructed, 

 and second, the best distribution for simplifying the resulting 

 motion of the string. The open space at the centre occupied 

 by the microscope and the ends of the string which project 

 beyond the field are not specifically considered, as it will bo 

 apparent how these irregularities may be allowed for in an 

 approximate way when the data are known. 



Let PQ, fig. 1, be the element of any string of length els, 



Fig. 1. 



T ( +^T ( 



and confined to a plane. The element is acted upon by the 

 tension T a at P and the opposite tension Ti+dT x at Q, and 

 the resultant of all external forces Fds in the plane of the 

 string acting at any angle %. 



If the element of the string is at rest these three forces are 

 in equilibrium, and resolving along the tangent, we find 



5+Fcos X = 0, (1) 



and along the normal 



^-Fsin X = 0, ...... (2) 



ds 

 where r = -j n = the radius of curvature at P. 

 du 



