322 Prof. S. P. Thompson. On the [Feb. 14, 



or, writing n for the ratio r/l, 



j_m| n — sin _ w-fsin0 1 

 ~T 2 1(>2 + 1 -2n sin 0)* (»2 + l + 2wsin 0)*J ' ^ 



which for brevity we may write 



P 



where P= W -sin0 ^ + sin0 



(w s + 1 - 2?i sin 0)f O 3 + 1 + 2n sin 0)1 



It is obvious from the mere form of this expression that it is not 

 accurate to assume that a scale of equal degrees will represent equal 

 increments of the magnetic force ; and that no scale ordinarily used 

 in galvanometers or magnetometers, such as a scale of sines or of 

 tangents, will adequately serve the purpose. It is, therefore, necessary 

 to investigate the formula to ascertain how it can be made available 

 for the graduation of a scale for the instrument. 



The expression is an awkward one for numerical calculation, 

 owing to the fractional exponents of the denominators ; but the 

 process of calculation can be somewhat shortened by taking n as a 

 simple integer, and the successive values of such that the sines are 

 simple decimals. In the following tables will be found the values of 

 the function P, calculated for several cases, the angles being in all 

 cases expressed in degrees and decimals of degrees, not in degrees, 

 minutes, and seconds. The object in calculating these values was to 

 ascertain the proportions of r and I, which would enable the scale 

 readings of the compensator to be used without the labour of con- 

 structing arbitrary interpolation tables for the calibration of the 

 readings. 



Table 1.-0=1.) 



Angle. 



Nat. sine. 



P calculated. 



o 















5-739 



o-i 



-0355 



11 -538 



0-2 



-0726 



17 -457 



0-3 



-1195 



23 -578 



0-4 



0-1576 



30 -000 



0-5 



-2113 



36 '869 



0-6 



-2794 



44-427 



0-7 



-3741 



53 -130 



0-8 



-5271 



64 -158 



0-9 



0-8618 



71 -805 



0-95 



1 -3280 



90 -000 



1-0 



00 



