68 MR EDWARD SANG ON THE MOTION OF A HEAVY BODY 



22. The periodic time of a body descending from C is deducible from that of 

 one which has the velocity belonging to a descent from A ; that again is deducible 

 from the motion of a body supposed to have fallen from A v and so on. Now, the 

 distances NA, NA 1? NA 2 , &c, increase with greater and greater rapidity as we 

 proceed, so that after a few terms NA may become enormously great as compared 

 with NZ. But EZ is proportional to the velocity at Z, while NE is proportional 

 to that at N ; and the ratio of NE to EZ must approach nearer and nearer to a 

 ratio of equality as the point A rises ; so that when NA is very great, the velocity 

 becomes almost uniform, and the periodic time of the motion becomes the 

 quotient of the circumference by that velocity. In this way, the study of the 

 law of this progression may conduct us to a knowledge of the periodic time of the 

 motion of 7. 



23. Analogously the distance NB is deduced from NC, from NB again we may 

 deduce NB 1S thence NB 2 , and so on; and in this manner, we may reduce the 

 question of the time of descent in the arc KN, to that of the time of ascent in 

 an excessively minute arc. 



24. Attending first to the case of oscillatory motion, let it be proposed to 

 compute the periodic time of a body having its velocity due to a descent from 

 the level of D. 



For this purpose, let us put B for the angle NZK measured by half of the 



extreme arc NK, and B a for the angle NZF measured by half of the arc NF ; 



then 



NK = NZ . sin B , KZ = NZ . cos B ; 



but 



NZ + ZK : NZ - ZK : : NZ : NF , 

 wherefore 



1 + cos B : 1 — cos B : : (cos £ B ) 2 : (sin $ B ) 2 : : 1 : sin Bj 

 and 



sin B 1 = (tan J B ) 2 . 



But it has been shown in article 20 that the times of descent from K and 

 from F, there marked by the symbols T s and T^ are in the ratio 2\/(NC . NE) : NE 

 or of 2\/NC : a/NE. It is now convenient to indicate these times by the charac- 

 ters T sin. B and time B x , so that the above proportion may be written 



\/NC : a/NE 

 2NI : NI + IZ 



2NZ : NZ + ZK 

 2 : 1 + cos B 



1 : (cos£B ) 2 



(sec J B ) 2 : 1 



so that the time of descent from K is 



time B = time B 1 . (sec ^ B ) 2 . 



time B : time B 2 



