454 Ona new Theoretical Determination of the Velocity of Sound. 
same, and equal to 
hm. {12 fl(h) +22. fl(Qh) + 3% £(Bh) +... HL 
From this formula we learn that every slice produces a term in 
the expression for the velocity, so that there are as many terms 
in the expression for the velocity as there are slices within the 
radius of the sphere of action of any one particle. 
6. From arts. 2 and 8 it is evident that the whole force exerted 
upon a particle of the slice F by all the slices on one side of F 
is mf(h) + mf(2h) +mf(3h) + .: +; and this is therefore the force 
which we must suppose concentrated in the slice B, and an equal 
force in the slice G, on the hypothesis of continuity. Hence if 
mE (hi) =mf (h)-+mf (2h) +mf (Bh) +... 
mE" (ht) =mf'"(h) + mf" (Qh) + mf" (BA) + ++ +4 
and the equation of motion will be, for the case of continuity, 
Dj 2,= mF" (h) . (vp-y — 22, + X41): 
From this we obtain, as before, the expression for the velocity of 
transmission, 
vel. =h Vm. F'(h). 
But in this case we know the velocity of transmission is / p, 
the velocity determined by Newton, 
v VMp=hvm. Ph) 
=h W/m. {f'(h) +f'(2h) +f'(8h) +...}7. 
7. Eliminating m between this equation and that of art. 5, 
there results, finally, 
ay. {ite +22, f'(2h) +37. f'(3h) +... le 
NF) EB) te PED) tries hs 
Now the numerator of this fraction is of necessity larger than 
the denominator ; and therefore, on the very face of it, this ex- 
pression indicates that the actual velocity of sound is greater 
than was found by Newton. It remains to determine the value 
of this expression. 
8. We assume that f(z), and therefore also f'(z), is some 
simple inverse power of z, That power in the case of f'(z) must 
then 
be greater than 3; for if f'(z) be equal to then the expres- 
sion in art. 5 gives the velocity of transmission 
20" CREE C bs 
hn. Vaataptgpt} 5 
which is known to be infinite. The lowest possible value of the 
