312 Dr. Roget on the 



of the acting poles ; which distances, in the case before us, 

 are the lines C N and C S. But that part of each force which 

 is effective in producing rotation, is, by the resolution of 

 forces, as the sine of the angle which the direction of the 

 force makes with the radius of rotation. Taking both these 

 circumstances into account, the rotatory forces exerted by the 

 two poles are to one another in a ratio compounded of the 

 sines of the angles N C T and S C T, and of the recipro- 

 cals of the squares of C N and C S. For the convenience of 

 notation, let these rotatory forces be denoted respectively by 

 the letters R and r. 



Let C N = n 

 CS = s 



The angle NCT = v 

 The angle S C T = a 

 The length of the magnet, or N S = m. 

 The portion of the produced axis S T intercepted between S 



and the line C T = x 

 From the points N and S, let fall upon C T 

 the perpendiculars N P = p 

 and S Q = q. 



The triangles N P T and S Q T being similar 

 p : q : : m + x : x 



Also sin v = ; and sin a -=L JL : 

 n s 



mi r> sin v sin a p q m + x x 



inen, \\ i T i 1 '. '. '. ; -ii i . i 



n* * a n 3 s 3 n 3 s 3 



But in the present case R = r ; therefore m + x = \ 



n 3 s 3 



hence n 3 s 3 : s 3 : : m : x. 



That is, C N 3 - C S 3 : C S3 : : N S : S T. 

 Hence, in order to determine geometrically the point T, in 

 the axis N S produced, and thereby the direction of the line 

 C T, which is the position of equilibrium for the infinitely 

 short needle C, and the tangent to the magnetic curve at that 

 point, we must take on that axis a distance S T such that it 



