72 Prof. E. H. Barton on Spherical 
behind over that due to pressure, p + dp>. in front, or 
P dy dz—(p + -^ dx) du dz = — J' dx dy dz. 
Hence, equating, we find — dp dx=pdu/dt, or 
ldp__ du 
p dx ~ dt W 
But p is some function of p. and, whatever the form of 
the curve coordinating p and p. the small portion with which 
we are concerned may be considered straight. Hence, for 
our small vibrations, we can write 
dp/dp=a* 3 some constant (4) 
We can easily see. by the method of dimensions, that a in (4) 
is of the nature of a velocity. For the left side of (4) is 
dimensionaily 
AILT--L - - 
- — -. rT _ 3 = L' : T~- = (velocity)-, 
in which M, L. and T represent mass, length, and time re- 
spectively. From (4) we get 
dp a 2 dp a-p-ds , , ._. 
-*- = -— — p — . =a-ds nearlv. . . (o) 
P P P; <1 + ') " 
It should be noted here that we are not entitled to inte- 
grate (4) and draw from such result any conclusions about 
the general relation between p and p. On the contrary, the 
relation between them must be determined independently. 
and the general value of dp/dp = a' 2 derived from it. 
Substituting (5) in (3). we obtain 
a*ds/dx=—du/dt (6) 
Whence, for the other axes, we have by symmetry of 
notation 
cFdsjdy^-dc.dt, (7) 
and *ds d:=-dw / dt (8) 
These three constitute the required equations of motion. 
The Differential Equation. — We have now to derive from 
equations (2), (6), (7). and (8) the partial differential 
equation. We see that in addition to s, x, y. c, and t. the 
only variables we wish to retain, the above equations involve 
also u. c. and w. These last three must accordingly be 
