60 



MR EDWARD SANG ON THE MOTION OF A HEAVY BODY 



from N with some given velocity V, it will ascend along the circumference, losing 



velocity as it rises. When the initial velocity 

 exceeds that which is due to a descent along the 

 diameter ZN, the body will rise to the zenith- 

 point Z, and will proceed onwards to descend 

 along the other semicircumference ; after that 

 it will continue (all resistance being supposed 

 away) to repeat revolution after revolution. 

 But when the initial velocity is less than that 

 which is due to a descent along ZN, the heavy 

 body will have lost the whole of its velocity at 

 some point below Z, from that point it will de- 

 scend again to N, pass to the other side, on 

 which it will reach to the same height, and 

 thence descending it will continue to oscillate 

 as in the familiar example of a pendulum. 

 There are thus two distinct cases of circular motion, viz., the continuous and the 

 oscillatory. 



3. These two cases may be connected in the following manner : — 

 Let us suppose that a heavy body a, has been projected at N, with a velocity 

 due to its descent from some point A beyond Z, and that it has now reached to the 

 point marked a. Having drawn the horizontal line aG, we see that its velocity 

 at the point a is that which is due to a fall through the distance AG ; so that if 

 we put V A for the initial velocity, and v a for the velocity at the point a, we 

 must have the proportion 



Va 2 : v a 2 : : NA : AG : : NZ . NA : NZ . AG 

 : : NZ . NA : NZ . NA-NZ . NG . 



Through Z draw the horizontal line ZE, and make it a mean proportional be- 

 tween NZ and ZA ; join NE, EA, then the trigons NZE, NEA are similar, so that 

 NE is a mean proportional between NZ and NA, wherefore the above analogy 

 may be written,— 



V A * 



2 : :NE 2 : NE--Na 2 



4. F being the intersection of NE with the circumference of the circle, draw 

 FB horizontally, then the five lines NA, NE. NZ, NF, and NB, are in continued 

 proportion ; so that NA : NZ : : NZ : NB. 



If a second body j3 be projected from N, with a velocity due to a descent from 

 B, it will rise along the curve only to the point F, its velocity there being entirely 

 exhausted. The greatest distance, then, which (3 can reach from N, viz., NF, is 

 to the greatest distance to which a can attain, viz., NZ, in the ratio of NE to NA. 

 Let us take an intermediate point (3 to correspond with a, by making Na : N/3 



