Page:Newton's Principia (1846).djvu/370

and in their return equal to EG + ln. But εγ is the breadth or expansion of the part EG of the medium in the place εγ; 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 + ln or EG + LN to EG. Therefore since LN is to KH as IM to the radius OP, and KH to EG as the circumference PHShP 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 æquo, LN to EG as IM to V; the expansion of the part EG, or of the physical point F in the place εγ, to the mean expansion of the same part in its first place EG, will be as V - IM to V in going, and as V + im to V in its return. Hence the elastic force of the point P in the place εγ to its mean elastic force in the place EG is as $\scriptstyle {\frac {1}{V-IM}}$ to $\scriptstyle {\frac {1}{V}}$ in its going, and $\scriptstyle {\frac {1}{V+im}}$ to $\scriptstyle {\frac {1}{V}}$ in its return. And by the same reasoning the elastic forces of the physical points E and G in going are as $\scriptstyle {\frac {1}{V-HL}}$ and $\scriptstyle {\frac {1}{V-KN}}$ to $\scriptstyle {\frac {1}{V}}$; and the difference of the forces to the mean elastic force of the medium as $\scriptstyle {\frac {HL-KN}{VV-V\times HL-V\times KN+HL\times KN}}$ to $\scriptstyle {\frac {1}{V}}$; that is, as $\scriptstyle {\frac {HL-KN}{VV}}$ to $\scriptstyle {\frac {1}{V}}$, or as HL - KN to V; if we suppose (by reason of the very short extent of the vibrations) HL and KN to be indefinitely less than the quantity V. Therefore since the quantity V is given, the difference of the forces is as HL - KN; that is (because HL - KN is proportional to HK, and OM to OI or OP; and because HK and OP are given) as OM; that is, if Ff be bisected in Ω, as Ωϕ. And for the same reason the difference of the elastic forces of the physical points ε and γ, in the return of the physical lineola εγ, is as Ωϕ. But that difference (that is, the excess of the elastic force of the point ε above the elastic force of the point γ) is the very force by which the intervening physical lineola εγ of the medium is accelerated in going, and retarded in returning; and therefore the accelerative force of the physical lineola εγ is as its distance from Ω, the middle place of the vibration. Therefore (by Prop. XXXVIII, Book I) the time is rightly expounded by the arc PI; and the linear part of the medium εγ 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 εγ as soon as it returns to its first place is at rest; neither will it move again, unless it