ALONG THE CIRCUMFERENCE OF A CIRCLE. 



71 



logarithm of the ratio of the time of oscillation in a semicircumference to that in 

 a small arc is thus 



Log sec \ B = 0-034 3847 

 Log sec iB x = 1 6160 



Log sec \ B 2 



30 



0036 0037 x 2 



1180311 



0-072 0074 



In other words, the time of oscillation in a semi-circumference is to that for a 

 minute arc as 72 to 61 very nearly. 



When the arc is nearly a whole circumference, the steps of the progression are 

 more numerous, and the computation may, with advantage, be systematically 

 arranged. I subjoin the work for an arc of 320°, with logarithms to ten places. 



n 



JB„ 



2 log sec ^ B„ 



2 log tan £ B n 



B n+1 







4°0 00 00000 



•23149 20670 



9-84762 70684 



44 45 21-339 



1 



22 22 40-669 



6800 53042 



922920 47378 



9 45 34-359 



2 



4 52 47180 



315 40158 



7-86265 75152 



25 03-441 



3 



12 31-721 



57684 



5-16071 41918 



2-986 



4 



1 493 







=log 200750740 





•30265 71554 = 



Thus, even in this extreme case, four terms of the progression are sufficient. 



These examples show that for all cases in ordinary clock-work, or in experi- 

 ments on the length of the pendulum, the time of oscillation is sufficiently well 

 represented by the formula 



timeB = 7r N /(i).(seciB) 2 



B being one- fourth part of the entire arc of oscillation. Hence the number of 

 beats per day is proportional to the square of the cosine of the eighth part of the 

 arc, and the daily retardation to the square of the sine of the same eighth part. 



29. From the periodic time of an oscillation that of the conjugate continuous 

 motion may be readily deduced ; and thus, so far as the entire motions are con- 

 cerned, the theory may be said to be complete. The investigation of the time at 

 which the moving body arrives at any proposed point in its path is carried on by 

 the application of the principle just explained ; but as it is of comparatively little 

 interest, and, at the same time, more tedious, I shall not, for the present, go into 

 its details. 



VOL. XXIV. PART I. U 



