364 THE MATHEMATICAL PRINCIPLES [BOOK II. and in their return equal to EG + In. But ey is the breadth or ex pansion of the part EG of the medium in the place ey ; and therefore the expansion of that part in its going is to its mean expansion as EG — LN to EG; and in its return, as EG -f In or EG + LN to EG. Therefore since LN is to KH as IM to the radius OP, and KH to EG as the circumference PHSAP to BC ; that is, if we put V for the radius of a circle whose circumference is equal to BC the interval of the pulses, as OP to V • and, ex cequo, LN to EG as IM to V ; the expansion of the part EG, or of the physical point F in the place ey, to the mean ex pansion of the same part in its first place EG, will be as V — IM to V in going, and as V -f im to V in its return. Hence the elastic force of the point P in the place ey to its mean elastic force in the place EG is as 11. 11, v fivf *° v m 1^s Somo> an. And for the same reason the difference of the elastic forces of the physical points e and y, in the return of the physical lineola ey, is as ftr/>. But that dif ference (that is, the excess of the elastic force of the point e above the elastic force of the point y) is the very force by which the intervening phy sical lineola ey of the medium is accelerated in going, and retarded in re turning ; and therefore the accelerative force of the physical lineola ey is as its distance from ft, the middle place of the vibration. Therefore (by Prop. XXXVIII, Book 1) the time is rightly expounded by the arc PI ; and the linear part of the medium sy is moved according to the law abovementioned, that is, according to the law of a pendulum oscillating ; and the case is the same of all the linear parts of which the whole medium is compounded. Q,.E.D. COR. Hence it appears that the number of the pulses propagated is the same with the number of the vibrations of the tremulous body, and is not multiplied in their progress. For the physical lineola ey as soon as it returns to its first place is at rest ; neither will it move again, unless ii