396 Intelligence and Miscellaneous Articles. 
time ab, then is torn off and jumps back to its first position during 
the time &c, till the bow again gets hold of it. 
With these principal vibrations smaller vibrations are compounded, 
the nature of which I can define accurately only in the case where 
the bow touches a point whose distance from the nearer end of the 
string is > - 7 &c. of its whole length, or generally — if mis a 
whole number. Because the point where the bow is applied is not 
moved by any vibration belonging to the mth, 2mth, &c. harmonic, 
_ it is quite indifferent for the motion of that point, and for the impulses 
exerted by the bow upon the string, whether vibrations corresponding 
to the mth harmonic exist or not. Th. Young has proved that if 
we excite the vibrations of a string by bending it with the finger, as 
in the harp, or hit it with a single stroke, as in the piano, in the 
ensuing motion all those harmonics are wanting which have a node 
in the touched point. I therefore concluded also that the bow 
cannot excite those harmonics which have a node at the point where 
it is applied; and I found, indeed, that if this point is distant 4 from 
m 
the end, the ear does not hear the mth harmonic sound, although it 
distinguishes very well all the other harmonics. Therefore, in the 
equation (1), all those members of the sum will be wanting in which 
n is equal to m, or 2m, or 3m, &c.. ‘These members, taken together, 
constitute a vibration of the string with m vibrating divisions. Every 
such division performs the same form of vibration we have described 
as the principal vibration of the whole string. ‘These small vibra- 
tions must be subtracted from the principal vibration of the whole 
string for getting its actual vibration. Curves constructed according 
to this thecretical view represent very well the really observed curves. 
If m=6 and the observed point is distant, | from the other end of the 
string, the motion is represented in fig. 10. Near the end of the 
string, where the bow is commonly applied by players, the nodes of 
different harmonics are very near to each other, so that the bow is 
nearly always at, or at least very near to, the place of a node. 
Striking in the middle between two nodes, I could not get a curve 
sufficiently constant for my observations. If I strike very near the 
end, the sound changes often between the fundamental and the 
second or third harmonic, which is indicated by gradual corresponding 
alterations of the curve.—From the Proceedings of the Glasgow Philo- 
sophical Society for Dec. 19, 1860. 
ON CLAIRAUT’S THEOREM. BY PROF, HENNESSY, F.R.S. 
Laplace has shown that this theorem follows, whatever may be the 
density of the interior parts of the earth, provided it consists of similar 
concentric strata, and that the form of the outer stratum is ellipsoidal. 
In the ‘ Philosophical Transactions’ for 1826, Mr. Airy (the present 
