1887.] Equations to the Motion of Perforated Solids. 121 
From (2) we see that the component impulse corresponding to 
u,, Which must be applied to S, in order to keep it at rest, when 
the cyclic motion is generated by the application of proper impulses 
to the barriers of all the solids is — p | | dd,'/dn. > (xdo); and 
w 
therefore by (3) the generalised component of momentum X, cor- 
responding to u, of the cyclic motion when all the solids are reduced 
to rest, is 
oe d if ihe! 
¥,=£-p|| PS (edo) = p|[ ¥,(eldo) - | Pt 5 (edo)...(4), 
whence X= - BEN Nc edie oesnereet a enieomegnet es (5): 
Similarly b= - +H, 
where Nap ies > («do), 6) 
and rN, = 2A, =p Sf >, [« (ny—mz) de] 
3. We must now obtain an expression for the modified 
Lagrangian function. 
Let the coordinates of a dynamical system be divided into two 
groups 0 and y, the latter of which does not enter into the 
expression for the energy of the system. Then it is known from 
the theory of ignoration of coordinates that, 
dT 
ae CONS Cy ype en ssrostisasncs Nose cesses @ 
and that if the velocities y be eliminated by means of the system 
of equations of (7) in the type, the resulting expression for 7’ will 
be of the form 
The quantities @, 8, & are defined as follows; let the original 
expression for 7’ be 
2T = (00)0 + 2 (00,) 60, +... 2(Oy) Ox +... (yy) H+ AKAIKE 
also let P= (6x) 6+ (6,y) 6, aa 
Ee = (Ox,) A+ (CAA) 0, a5 
Then it is shewn in Thomson and Tait, Vol. 1. Part 1. pp. 
320—323 (writing « for C) that 
X = (we) (« — P) + (e,)(e, — ia Meth: (10) 
X, = («K,)(« — P) + («,«,)(«, — P,)+ 
