THE EEV. S. EAEHSHAW ON THE MATHEMATICAL THEOEY OP SOUND. 135 
4. We have now to express these results in terms of the original genesis of the 
motion. Let us suppose the motion generated by a piston pushed forwards in the tube 
in a given manner. Let the piston at the time T (having the same origin as t) be at 
the distance Y from the plane of reference, and moving forwards with the velocity U ; 
and by K denote the density of the air in contact with the piston at that moment. For 
all particles in contact with the piston a^=0 (we suppose the piston to commence its 
motion at the origin of nc). Then since at the time T the particles in contact with the 
piston are within the limits of the wave, equations (6.) and (7.) must be satisfied; 
4L 
Y=+\///-logy.T+^(a')] 
0 =+\/ i 
In these equations and at present we have not sufficiently connected the two 
systems of equations (7.) and (8.). We shall further connect them by assuming E=g’, 
which gives a' = a; the effect of which assumption is to limit the meaning of T, Y, U 
as follows : — 
T is the time of genesis of the density § which at the time t has been trans- 
mitted to the place denoted by y ; 
Y is the place where the density was generated ; 
U is the velocity of the piston when ^ was generated by it. 
Y e may now write a for a', and then eliminate a, (p(a), and <p'{a) between the four 
equations (7.), (8.). By this means we obtain 
y=Y+{V+^-i;.)(t-T) (9.) 
,T=+y^s'^'^(j>-T) (10.) 
_U 
( 11 .) 
5. By these equations the state of a wave at any moment is connected with its genesis ; 
and they contain in fact the complete solution of the problem of every kind of motion, 
in a tube, which can be generated by a piston. 
6. From (11.) it appears that u = \] ; that is, that the particle-velocity generated by 
the piston is transmitted through the medium without suffering any alteration. The 
same equation (11.) shows that between the density and the velocity there is an inva- 
riable relation, which is independent of the law of original genesis of the motion ; so 
that in the same wave, or in different waves, wherever there is the same density, there 
will also be the same velocity. 
7. One of the most obvious facts on looking at the equations just found is, that for 
the same genesis there are two values of x, two of y, and two of The signification of 
Number of the Magazine it also appears that Professor De Moegan had discovered and communicated to 
the Astronomer Eoyal two particular forms of the function F ; without perceiving, however, that a slight 
generalization of his results would put him in the way to the integral expressed by the equations (5.). 
