232 PROCEEDINGS OF THE AMERICAN ACADEMY 



Each parabola was constructed from two series of measurements, in 

 both of which the relative directions of the current in the outer and 

 inner coils were unchanged, each series being represented by a branch 

 of the parabola. 



It was next necessary to find the law which the dynamometer 

 would follow for alternating currents. 



Taking the forces acting along any line passing through the centre 

 of the coil, we have for the action between the coils a force propor- 

 tional to the square of the current, and for the action of the earth's 

 magnetism on the suspended coil a force proportional to the current. 

 These two forces are balanced by torsion. Expressed in the form of 

 an equation, this gives aO 2 + f3G = y, where G is current and a, /?, y 

 constants for that position. To obtain the same deflection we might 

 have used a negative current, G 1 , which may be expressed as — nC; 

 then an 2 G 2 — (3 n G = y ; from these two equations it follows that 

 aC 2 n — y, that is, eliminating the earth's action, a steady current of 

 G*Jn would produce equilibrium in this position. But, as G 's/n 

 is a mean proportional between the arithmetical values of the two 

 currents which produce equilibrium in this position, we have simply 

 to find the mean proportional for each position, and construct a new 

 curve. Solving equations (1) and (2) we get 



1.297 



c = ^p ± |/^zy 2 ± 0il 345-. 



A mean proportional between the values of G for (1) is -0.134.aS'; 

 for (2) it is +0.134 S. Therefore, the equations for a direct current 

 when the inner coil is free from the earth's effect are 



(72 = -0.134 S, (3) 



and 



G 2 = +0.134 S. (4) 



(See Figure 2). In the case of an alternating current, we have a con- 

 tinuous variation in strength ; but if we understand by G 2 the mean 

 value of the square of the current, and not the square of its (arith- 

 metically) mean value, which is nearly 20 per cent smaller, when 

 the current variations are of a simple harmonic character, we may 

 use equations (3) and (4) to express variations of current and deflec- 

 tion for alternating currents. In changing from (1) and (2) to (3) 

 and (4), we have made a change in the axes only, the curves being 

 identical. 



