Summing the series, we easily find that the vibration on the surface of the enveloping sphere

−1·708 P3 (cos @) + 2·600 P ̧ (cos @) + ...]eket (11)

Ø)

We have seen that when the vibration on the surface of the enveloping sphere is

ΣAP (cos 0). eiket,

the velocity potential of the wave-disturbance is

(a+b)2_ik(et-r+a+b) 、 AnPn (cos 0)

e

Fn(ik.a+b) n(ikr).

Now when r is large, f(ikr) = 1, so that the factor on which the relative intensities in various directions depend is

Thus if we put this quantity = F+iG, the intensity of the vibrations in various directions is measured by F2 + G2.

The distribution of intensities in different directions round the spheres will be influenced to a considerable extent by the value of the wave-length chosen. If we take k(a+b)=2, the wave-length is 37 inches, and if we take k(a+b)=3, the wave-length is 27 inches. From the expression (1) for the wave-motion produced by a single sphere undergoing an instantaneous change of velocity, it is seen that the wavelength to be chosen is of the same order as the circumference of the sphere. From this, it appears that for a system of two spheres whose radii are 1 inch and 2 inches respectively, the wave-length to be chosen should be some value intermediate between 2 and 4π, probably nearer 2π than 4π; for, in the actual case of impact, the smaller ball which would undergo by far the greater change in velocity would probably influence the character of the motion to a greater extent than the larger sphere. At the same time, it must not be forgotten' that the analogy between the cases of impact and of periodic motion cannot be pushed very far, inasmuch as the fluid motion due to impact is undoubtedly

of different character in different directions, and not all throughout the same as in the periodic case.

Now taking k(a+b)=2, we find (neglecting a constant factor)

F=0992 P1 (cos )+2840 P2 (cos 0)—·0354 P2 (cos 0)

1

2

G='0496 P, (cos 0)-4040 P2 (cos 0)-0177 P3 (cos 0)

1

2

The values of F and G for different directions have been calculated and are shown in the following table :

Now taking k(a+b)=3, we find (neglecting a constant

factor)

F= 105 P1 (cos ) + 1.060 P2 (cos )+016 P2 (cos 0)

2

G122 P, (cos 0)+186 P3 (cos 0)-024 P1 (cos 0)

4

(13)

The values in different directions have been calculated from these expressions and are shown in the following table:

The values of F2+ G2 shown in Tables I. and II., have been plotted in polar coordinates in figs. 3 and 4 (Pl. IV.). It is seen that in both cases the intensity in the direction of the larger ball is greater than in the direction of the smaller ball. The asymmetry is more marked when k(a+b) has the larger value.

The intensity of the sound in different directions due to the impact of two spheres of wood of diameters 3 inches and 1 inches respectively has been measured with the ballistic phonometer and is shown in fig. 5. It is seen that this curve is intermediate in form between those shown in fig. 3 and fig. 4, exactly as anticipated. as anticipated. The agreement between. theory and experiment is thus very striking in this case.

3. Two spheres of the same diameter but of
different materials.

We have seen in the preceding section that in the expressions for F and G for two spheres of the same material but of unequal diameters, the terms containing the zonal harmonic of the second order P2 (cos e) usually preponderate, and that the intensity diagram is, accordingly, a curve which consists of four loops. A different result is obtained in the case of two spheres of the same diameter but of markedly unequal densities. The zonal harmonic of the first order preponderates in this case, and the intensity diagram is a curve consisting of only two loops. To obtain this result theoretically, we have to proceed on exactly the same lines as in the preceding pages.

Taking a=1 inch and b=1 inch, we easily find from the expressions (7) and (8) that

Summing the series, we find that the vibrations on the surface of the enveloping sphere, namely

[(U-U1) × 2254 P1(cos ) + (U2+U ̧) ×·3550 P2(cos 0)

α

+(U2−U2) ×·3645 P3(cos Ø) + (U2+U¿) × ·3080 P1(cos 0)

a

If the ball of radius b is four times heavier than the one of radius a, we have

U1 = 4U.

So that the vibration on the surface of the enveloping sphere is proportional to the expression

•6762 P1(cos ) + 1·7750 P2(cos ) + 1·0935 P ̧(cos 0)

+1-5400 P4(cos 0)+6975 Ps(cos 0)+ &c.

Now taking k(a+b)=1, which will give a wave-length equal to the circumference of the enveloping sphere, we get (neglecting a constant factor)

F= 13524 P,(cos 0)-04987 P2(cos 0)-0074 P(cos 0) +0007 P (cos 0)+ &c.

4

G0676 P1(cos )-0798 P2(cos )+0047 P2(cos 0)

The values of F and G, and of F2+ G2 in different directions. obtained from the preceding expressions are shown in Table III.

The values of (F2+G2) shown in Table III. have been plotted in polar coordinates and are shown in fig. 6. It is seen that the maximum intensity in the direction of the heavier ball is greater than that in the direction of the lighter one.

The experimental curve of intensity of sound due to impact of a sphere of wood, diameter 24 inches, with a billiard ball of nearly the same size is shown in fig. 7. It is found