1884.] Theory of the Magnetic Balance of Hughes. 325 



Table VIL— 0=10.) 



Angle. 



Nat. sine. 



P calculated. 



o 















5 -739 



-1 



-000390 



11 "538 



0*2 



0-000778 



17 -457 



0*3 



-001169 



23 -578 



"4 



-001505 



\J \J\J -L. <J\JtJ 



30 -000 



0-5 



-001966 



36 -869 



6 



-002374 



44 -427 



0-7 



-002784 



53 -130 



0-8 



-003207 



64-158 



0-9 



-003638 



71 "805 



0-95 



-003858 



90 -000 



1-0 



-004090 



The calculated values of P are plotted out in the two sheets of 

 curves (figs. 2 and 3) accompanying the tables. The curves for w=4 

 and n=5 are given twice, being drawn again with enlarged ordinates 

 to show on a larger scale their close approximation to straight lines 

 up to about 50°. 



[3.] From the foregoing tables, and from the curves appended, 

 several conclusions may be immediately drawn with respect to the 

 design of the magnetic balance. A simple inspection of the curves 

 will show that they belong to a family in which, in general, there is a 

 concavity near the origin, and a convexity as the limiting value is 

 approached at the point corresponding to the 90° position of the com- 

 pensator. Those curves for which n=3, or less than 3, show the 

 convexity very markedly. In those for values of n higher than 7 

 (only one has been drawn, namely, that for n=10) the convexity of 

 the upper portion asserts itself. The curve for n=10 approximates 

 very nearly to a curve of sines, as indeed might be suspected from the 

 equation. 



For those values of n which lie between 4 and 6 inclusive, the first 

 half of the curve is very nearly a straight line, so nearly so that for 

 the curves w=4 and n—h the values of the ordinates do not differ 

 by 1 cent, from those which they would have for actual straight lines 

 lying along the mean slope of the respective curves as far as 45°. Tn 

 other words, for angles less than 45° the magnetic force exercised 

 along the axis of the balance by the compensator when it is turned is 

 proportional, within 1 per cent, of accuracy, to the angle through 

 which it has been turned, provided the distance of the compensator from 

 the needle be not less than four and not greater than five times the half- 

 length of the compensator. 



This result may be verified from equation (2) by finding what 



