406
THE THEORY OF SCREWS.
[366-
from Lagrange’s equations,
d (dT\ dT _ „
dt \dej ~ de; " P1V1 ’
d (dT\ dT
dt \df)J den'
These equations admit of a transformation by the aid of the identity
01
dT
de.'
dT
+ 0nde,;
Differentiating this equation by 0., we find
dT . d-T A d2T
de.' de.de.' de.dø,;
but
d (dT\ a d2T a d-T d2T A d2T • d2T
dt'déj de.2 de.de. dø.dén de.de. dé.dø,;
whence, by substitution
d fdT\ A d?T a d2T dT
dt\dej de.2 de.den de;
Hence when screw-chain co-ordinates are employed Lagrange’s equations
may be written in the form
a d2T -a d*T .a d2T „ / „ 1 dT \
de; de.de. de.den pv P.de;r
0 der + d2T 0 d^
dé.dén dé2dén n dén
■ „ 1 dT
Vn + —
Pn de,.
367. Generalization of the Eulerian Equations.
The equations just written can be further simplified by appropriate
choice of the screw-chains of reference. We have already assumed the
screw-chains of reference to be co-reciprocal. If, however, we select that
particular group which forms the principal screw-chains of inertia 357),
then every pair are conjugate screw-chains of inertia besides being reciprocal.
In this case T takes the form
T=M{u;e; + ... e,v + e;^-,+ ... e,; 1e;* + &c.
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