452 



MR EDWARD SANG ON THE MOTION OF A HEAVY BODY 



the whole circumference in the one case is equal to the time of an oscillation 

 in the other case. 



Farther, if we put C for half the exterior angle at c, that is, for the half 

 sum of A and B, we have 



AB 2 = a 2 + b 2 + 2ab cos 2C 



= a 2 + 2ab + b 2 - 4«& sin C 2 ; 



now 



2dC 

 dt 



= J2g . AB = J2g . J {(a + bf - ±ab sin C 2 } , or 



dc 

 di 



= ^W{(^) 2 -«6sinC 2 }, 



wherefore if we put C for the half sum of a and b, d for the mean proportional 



dC 

 between them, we have -=- = J2g . J(c 2 — d 2 sin C 2 ), an equation identic in 



form with that for the variation of A. Hence if we produce AC, draw CD, 

 fig. 10, bisecting the angle BCE, lay off CE a mean proportional between AC 

 and CB, and then inflect CD equal to the half sum of these same, the distance 

 QD, which is just half of AB, will represent the velocity with which the angle 

 ECD changes. At the same time CQ will represent the rate of change of the 

 angle at D, or 



dD 

 dt 



= J'2g V(d 2 -c 2 sinD 2 ), 



and thus we can obtain another pair of motions, one continuous, the other 

 alternate, synchronous with each other and with the preceding pair. 



Fig. 10. 



From this trigon CED we can, in the same way, that is, by making 



