of the string during the time the hammer is in contact is given by the following equations of Lord Rayleigh (Rayleigh, 'Theory of Sound,' vol. i. p. 204) :

and

- Mλ sin λa sin λb=p sin Al (frequency equation),

y= ΣP, sin λv sin λ‚a cos (cλ,t—er)

between x=0 and x=a,

y= Σ P, sin λ,(l — x) sin λ,b cos (cλ‚t — €r)

between a and x=l,

(1)

(2 a)

(26)

where P, e, are arbitrary constants, a and b are the two parts into which the string of total length is divided by the hammer or the load, c is the velocity of the wave along the string, M the mass of the hammer, and p the linear density of the string. As, however, the system is started into motion by initial velocity at aa by the hammer whose velocity just before contact is v, one set of arbitrary constants, say e,, will drop out and the equations will take the form: y= EP, sin λ, sin λ,(la) sin cλ,t.

(3 a)

(36)

between ra and x=l,

and thus at x=a

yoΣP, sin Ara sin A,b sin cλpt.

(3 e)

The arbitrary constants P,, etc. of the above series can be evaluated from the initial conditions either directly or with the help of Art. 101 of Lord Rayleigh's Theory of Sound,' vol. 1. The second method, which will be called Rayleigh's method, has been followed by Raman and Banerji. In the present paper it seems advisable to give both the methods.

Rayleigh's Method.

It follows directly from equation (7) of art. 101 referred to above (with present notations):

where M is the mass of the load, i. e. hammer, v its velocity before impact, and

Spud S

P,2 sin,a sin2 bdr

+pf'P?' sin3x,(1—2) sin2 \‚a.de. (5)

S

a

Taking the value of psu,2da as given in (5), we have from (4), after a number of transformations,

M (sin2 Ara + sin2 Ab)

(6)

1

Xb

which is different from that obtained by Raman and Banerji *.

Direct Method.

From equations (3a) and (3b) we have on differentiating and putting t=0,

(3)= Σ Ρ.λ. sin λ.æ sin a,b.

(7 a)

between x=0 and x=α,

(y)=EPA,c sin λ(1-x) sin A,a.

(76)

between xa and x=1,

and hence, using the normal property of the function, we

get

SP(y), sin λ,(l—x) dr = Prλ,ep sin λ,af" sin2 λ,(1 — a) dæ. (86)

Remembering now that (y)=0 except at x=a, we have the left-hand sides of equations (8 a) and (8b) equal to

S

a

* If, however, the Wad, i.e., the hammer, is taken as part of the string, then both the methods will give the same equation as that obtained by Raman and Banerji.

On adding (9a) and (96), we get

Mv=Prλrep [sin / S
do, a

+

0

sin λα

a

sin2 x da

S

sin λb sin3 \,(1-x)dx]. . (10) \,(l—æ)

arb

Now the above equation (6) gives the value of yo as an infinite series. And if X's in the different terms have simple ratios-in which case only is there great acoustical interest,the sum will represent a Fourier's series. It is well known to mathematicians (vide Hobson, Theory of Functions of a sin cart Real Variable, etc.,' p. 635 et seq.) that Σ

እr

λμ

between

- and is a non-uniformly convergent series representing a discontinuous curve. And therefore it cannot be further differentiated term by term. It can be easily seen. that the right-hand side of equation (6) is exactly of the same form as above except the factor*

of the string. Thus the value of expression (12) cannot

value of (12) therefore varies between 0 and

does not change sign. If we take any possible value (say k)

between the limits 0 and

M m- M

of the factor (12) for all

the terms of the series, equation (6) becomes

The right-hand side of equation (13) is convergent. But on differentiating term by term, the series thus obtained duo d2yo dt

becomes divergent. Thus or

dt2

cannot be obtained

by differentiating equation (6) as has been done by the previous writers (loc. cit.) to obtain the pressure of the hammer.

In conclusion we may quote the remark made by Raman and Banerji about such a divergent series :-". . . a difficulty arises in attempting to carry out a numerical summation of the series for all values of t, owing to the discontinuous nature of the function which the sum represents. difficulty may, however, be evaded," etc.

This

Note added in proof.-Further results confirming the above view have been obtained. They will be very shor ready for the press.

XXVI. X-Ray Analysis of Silver Aluminium Alloys. By Prof. A. F. WESTGREN and A. J. BRADLEY, M.Sc., Ph.D.*

[Plate IV.

HE silver aluminium system has been investigated by Petrenko †, who arrived at the equilibrium diagram reproduced in fig. 1. According to this, there should be

*Communicated by the Authors.

Zeitschrift f. anorg. Chemie, xlvi. p. 49 (1905).

two intermediate phases present in this system at ordinary temperatures, one corresponding to AggAl and the other being a "mixed crystal phase made up of Ag Al and Ag, Al. Both should be formed through transformation in the solid state.

In the following the former phase is called B' and the latter 7. The solid solution of aluminium in silver is denoted by a, and the aluminium phase by 8.

In order to make an X-ray analysis of the system, we have produced a series of alloys by melting together pure

Equilibrium diagram of the Ag-Al-system according to Petrenko.

silver with electrolytic aluminium in different proportions. The composition of the specimens was controlled by chemical determinations of both silver and aluminium. Small pieces of the alloys were filed or crushed into fine powder, which was recrystallized by heating in vacuo for some minutes to a temperature about 100°C. below the melting-point. From these powders, photograms were taken in a set of focussing cameras constructed by G. Phragmén.

From the series of photograms reproduced in Pl. IV. it is evident that Petrenko's statement concerning the number