1887.] Equations to the Motion of Perforated Solids. 123 
>i 
do dé 
d 
+ («—P) 0 (xx) + («, — P,) a (KK) + eee (15). 
Multiplying the system of equations of which (15) is the type 
by «, «,... respectively and adding we obtain 
d&dk  d& dk, 
Ale aD a di, Gato 
d&dP d&t dP, 
Dds COmn dic. a0) | 
Wigtaig, Wea Nae yo 
dé CTE Gs Ci ve 
Multiplying the equations of which (12) is the type by 8, 9.... 
respectively and adding, we obtain 
~; AS AS 
300) =P T+ PG. + 
pica) vaPiag y dst 
whence dé >(@8) =a 3 (Fa *) > (P76 : in) ’ 
GSE FNS gg U, 2 any 
therefore (16) becomes 74+ 7, — Ga >(©6) = 0, 
Td See Ue 
whence Apis ape de” db > (@6). 
We may now drop the symbol 0/d@ on the understanding that 
the momenta « are to be treated as constants, and Lagrange’s equa- 
tions become 
dS dO. ds dR idee dV 
Da Ge oe ee ea 
Since © and & do not contain 6, the modified function is 
D2 SOG BV ok tas (lm): 
If the velocities 0... be expressed in terms of new velocities w..., 
and ¥ be the new momentum corresponding to u after the w’s have 
been destroyed, it can easily be shewn that, . 
> (06) = & (Xu). 
For let 6=Aw+ Au, + A,u,+ 
then d6/dw=A, d6/du,=A,, &e. 
dO dL bs 
lso b Byer ne. un eee 
geoy Ale), (6 ol du du ‘ 
* In this term T is supposed to be expressed in terms of the velocities 
wu. , and x.... 
