238 
Proceedings of the Royal It'ldi Academy . 
ing pair of instantaneous screws. If A be reciprocal to Y, then B is 
reciprocal to X. 
A set of six impulsive screws, X,, &c., Xg, acting on a rigid body, 
can be chosen in an infinite number of ways, such that each instan- 
taneous screw {A]_, corresponding to X,, for example) is reciprocal to 
the five remaining impulsive screws (X3, &c., Xg). The six screws, 
Ax, &c., A^, are called conjugate screws of Idnetic energy. 
It can be shown that, in the case of a perfectly free body, one set 
of screws, A^, &c., A^^ can be chosen, which are conjugate screws, 
both of kinetic and potential energy. The six screws thus defined are 
called normal screws. If a body be displaced from a position of 
stable equilibrium by a twist about a normal screw, the body will 
continue to oscillate for ever, by twisting backwards and forwards 
along the screw. Whatever be the initial displacement of the body, 
it can be resolved into six twists about the six normal screws ; and 
whatever be the initial twist velocity, it can also be resolved into six 
twist velocities about the six normal screws. Whatever be the small 
oscillations, the movement is necessarily compounded of six twist 
oscillations about the six normal screws. 
In general, when a body has ^degrees of freedom, its small oscil- 
lations in the vicinity of a position of stable equilibrium are com- 
pounded of ^ twist oscillations about ^normal screws. 
The memoir further contains several general theorems with re- 
ference to the system of co-ordinate screws of freedom K and the re- 
ciprocal system ; a development of the principle of reciprocity ; an ac- 
count of the geometrical properties of the cylindroid ; and, finally, a 
detailed examination of the co-ordinate system appropriate to each 
degree of freedom. Figures representing models of the cylindroid and 
pectenoid illustrate the paper. 
XXYI. — Obseevatiot^s o?t Earl Stanhope's alleged Imperfections 
or THE Tfning-eore:. By Michael Donovan, Esq. 
[Read November 30th, 1871.] 
The instrument called the ''Tuning-fork," so well known to every 
musician, has long been used as the means by which the pitch of the 
various musical instruments used in a concert is to be determined, in 
order to attain perfect accordance. 
That it deserves this confidence was several years since called in 
question by Earl Stanhope, in his ''Principles of the Science of Tun- 
ing Instruments with fixed tones." The following are his words : — 
" There is, in practice, an objection to forks which is not generally 
known. It is this : out of a hundred forks, there is, perhaps, not one 
which has not a beating in it when it is struck. How, then, is it 
possible to tune an instrument accurately by means of forks which do 
not yield a pure or single sound ?" 
