454 



ME EDWARD SANG ON THE MOTION OF A HEAVY BODY 



Thus, when we have continued the precession so far as to make GI = IH, 

 the time in which the angle GFE has been generated can readily be found. 



In order to obtain a perfectly clear idea of this scheme of approximation, 

 let us draw through c, A, C, E, G, &c, lines perpendicular to CG, and inflect 

 to these Ad' = Ad ; CB' = CB, ED' = ED, &c. ; then the angles d', B', D', F at 

 the extremities of these are evidently the maximum values of d, B, D, F ; and 

 approach, toward the end of the series, more and more rapidly to a right 

 angle. 



Suppose that we wish to compute the time in which an arc of 6° is described, 

 when the entire arc of oscillation is 7° to each side of the vertical line, the 

 diameter of the circle being unit. Having made the angle B'Ad' = 3° . . 35', 

 since it is the angle at the circumference subtended by the arc of oscillation, 



and measured Ad' = J(-r), in this case unit, we draw d'c perpendicular to 



the horizontal line AB' ; and then construct on the other side the trigon cAd, 

 having Ad — Ad', and cdA = 3° ; after which the formation of the series pro- 

 ceeds as already described. 



The partial motion of the body through 6° of a total arc of 7° is now 

 synchronous with another motion through an arc 2B of a whole arc 2B', but in 

 a different circle ; and lastly, it is synchronous with a motion through 2H°, when 

 the body would just reach the zenith point of its circle. Hence the time would 



be expressed by I J2g = m . log tan (45° + \ H). Hence the following scheme 

 of calculation : — 



IH' 



cA = 

 Ad = 



•061 04854 

 1-000 00000 



8-785 6753 

 0-000 0000 



8-785 6753 

 9-668 1324 

 9-969 0427 

 9-999 7242 

 0-000 0000 



3 30 00 = 



27 00 26 = 

 68 37 21 = 

 87 57 30 = 

 90 00 00 = 



= r 



= B' 

 = Jy 



1-061 04854 



8-785 6753 



AC = 

 CB = 



•247 08000 

 •530 52427 



9-392 8376 

 9-724 7052 



•777 60427 



9-117 5428 



CE = 

 ED = 



•362 05235 

 •388 80213 



9-558 7714 

 9-589 7287 



•750 85448 



9-148 5001 





EG = 

 GF = 



•375 18888 

 •375 42724 



9-574 2500 

 9-574 5258 



= F 



i 



= li' 



•750 61612 



9-148 7758 



GI = 

 IH = 



•375 30806 

 •375 30806 



9-574 3879 

 9-574 3879 



A = 



7-099 44 



0-851 2242 









Here the data are Ad = 1 '000, d' = 3° 30' ; having written these in their 

 places, and also the logarithm of Ad in the second column, the log sine of d' 



