ELECTRO-MAGNETISM. 



17 



of the wire on that side of the needle ; tion required. This will appear from 

 but the direction of the motion produced the following demonstration : 

 is more or less oblique to the line con- //y<r. 29. 



necting the centre of the needle with ' s 



the wire. This direction may, in all 

 cases, be easily found by drawing the 



lines C * and C n (fig. 28), respectively /' ' 



perpendicular to WS and WN, and com- 

 Fig. 28. 



\ 



pleting the parallelogram C s a n ; of 

 which the diagonal, C a, will be the 

 direction of the resultant force acting 

 upon C. For the forces at S and N, 

 being inversely as the distances WS and 

 WN, are in the ratio of WN to WS, 

 which is equal to the ratio of the sines 

 of the opposite angles WSC to WNC of 

 the triangle WSN ; that is, in the ratio 

 of C s to C n, which are the actual sines 

 of those angles with the equal radii SC 

 and NC. The lines C* and Cn will, 

 therefore, correctly represent, both in 

 their directions and in their relative pro- 

 portions, the tangential forces in ques- 

 tion. 



(48.) The actions exerted between the 

 wire and the poles of the needle, are, as 

 we have seen, reciprocal ; the wire being 

 urged by a force equal in intensity, and 

 parallel in its direction, to that which 

 acts upon the centre of the needle ; 

 hence the determination of this resultant 

 force will also give us the measure and 

 direction of the resultant of the two 

 forces which act upon the wire. Thus 

 the needle SN, jig. 28, being urged by a 

 force represented by C a, the wire W 

 will, in like manner, be impelled by a 

 force represented by the line W w, equal 

 and parallel to C a, but having an oppo- 

 site direction. 



(49.) The direction of the force im- 

 pelling the wire by the joint action of 

 the poles of the needle, may be found 

 geometrically, by describing a circle 

 W6 Sr Na,fg. 29, which shall pass 

 through the position of the wire, and 

 also through the two poles ; for the dia- 

 meter W r of that circle will be the direc- 



Through S and N draw S r and N r, 

 respectively perpendicular to WS and 

 WN, and which will, of course, meet at 

 r, the extremity of the diameter W r; 

 and through W, draw W s and W n, 

 parallel respectively to r N and r S, 

 meeting them, when produced, in s and 

 n, and forming a parallelogram, of which 

 W r is the diagonal. The triangle W r s, 

 or its equal, Wrn, is similar to the 

 triangle WSN, because the angles WNS 

 and W r S, which subtend the same arc 

 W b S, are equal ; as also the angles 

 WSN and sWr, or its equal WrN, 

 which subtends the same arc WN. 

 The sides of these triangles are, there- 

 fore, proportional ; that is, 



WN : WS : : sr, or its equal Wn : 

 Ws. 



But the tangential forces impelling W 

 in the directions W n and W s, from the 

 actions of the poles S and N, are in- 

 versely as the lines WS and WN ; that 

 is, directly as WN to WS, and therefore 

 in the ratio of the line W n to the line 

 W*. These lines will, therefore, repre- 

 sent, in their magnitudes as well as in 

 their directions, the two tangential forces 

 by which W is impelled ; and conse- 

 quently the diagonal Wrof the parallel- 

 ogram of which they are the sides, or the 

 diameter of the circle, will represent the 

 direction of the resultant force in ques- 

 tion. 



C 



