62 MR EDWARD SANG ON THE MOTION OF A HEAVY BODY 



Now, if we suppose that the semicircumference NaZ is divided into a multitude of 

 minute portions, of which aa may be taken as one ; and if we divide the arc N/3F 

 into as many corresponding portions by making the chords N/3 always to the 

 chords Na in the constant ratio NZ to NE ; the time of describing each element 

 aa of the semicircumference NZ is to that of describing the corresponding 

 element f3(3' of the arc NF in the constant ratio NZ to NE ; and, consequently, 

 the time of describing any portion as Na must be to that of the describing the 

 corresponding portion NS in the same ratio ; and so also must be the periodic 

 times of the two motions. 



Hence, if we can discover the law of the motion in the arc NF, we shall be 

 able thence to deduce the law of the continuous motion due to the velocity 

 obtained by descent from the point A ; and contrariwise. 



For the sake of convenience, we shall call these two motions conjugate to 

 each other. 



8. It will conduce greatly to the clearness of our subsequent investigations to 

 introduce here another consideration. The time of describing the arc NF is 

 greater than that of describing the conjugate arc NZ in the ratio of NZ to NF; 

 the oscillatory motion will thus fall behind the continuous motion. Now, if we 

 were to suppose that the body (3 is acted on by gravitation of an intensity greater 

 than that which acts on a in the ratio duplicate of the ratio of the actual periodic 

 times, the two motions would be rendered alike. 



We shall then suppose that the gravitation acting on a, which we may desig- 

 nate by G„, is proportional to NZ, while the intensity of the gravitation acting on 

 /3, appropriately denoted by G^, is proportional to NA. And, as we have to do 

 with the subduplicate of the ratio of these intensities, we shall, for the sake of 

 additional convenience, put 



G« = NZ . NZ ; G^ = NZ . NA . 



9. The height from which a body has fallen being denoted by h, and the in- 

 tensity of gravitation being G, the velocity acquired is, according to the well- 

 known law of motion, proportional to V{Gh) ; we shall, therefore, put the general 

 formula for that velocity thus : — 



v = J(G . NZ . h) . 



10. On inserting the above value of G a in this general formula, and at the 

 same time making h = AG = NA — NG we have 



v a = J {NZ . NZ . NZ (NA - NG)} = NZ . V(NE 2 - Na 2 ) . 

 And similarly for the body /3 



vp = J {NZ . NA . NZ (NB - NH)[ = NE . J(NF 2 - N/3 2 ) . 



