670 
Proceedings of the Royal Society 
surface of fluid coinciding with 12 at any instant, let pressure be 
applied of constant value K per unit of area, over the whole area ; 
and at the same time let force (or force and couple) be applied to 
B equal and opposite to the resultant of this pressure supposed for 
a moment to act on a rigid material surface 12 rigidly connected 
with B. The “ motive” (that is to say, system of forces) consisting 
of the pressure K on the fluid surface, and force and couple B as 
just defined, constitutes the generalised component force corre- 
sponding to x [Thomson and Tait, § 313 (&)] ; for it does no work 
upon any motion of B or other bodies of the system if x is kept con- 
stant ; and if x varies work is done at the rate 
Kx per unit of time, 
whatever other motions or forces there may be in the system. 
Lastly, calling the density of the fluid unity, let k denote u circula- 
tion ” * [Y. M. § 60 (a)]f of the fluid in any circuit crossing j3 
once, and only once : it is this which constitutes the generalised 
component momentum relatively to x [Thomson and Tait, § 313 
(e)] ; for (Y. M. § 72) we have 
«=/„K *. • ■ • (4), 
if the system given at rest (or in any state of motion for which 
k — 0) be acted on by the motive K during time t.\ 
3. The kinetic energy T is, of course, necessarily a quadratic 
function of the generalised momentum-components, £, rj, ...k, k ... ; 
with coefficients generally functions of » J/, <p , but necessarily 
independent of x, ... ■ In consequence of this peculiarity it is 
convenient to put 
T = Q (f — olk — a 'k — &C., 7]- /3k- (3'k — &C., • • .) + ^ (k, k', . . .) (5), 
* Or fFds if F denote the tangential component of the absolute velocity of 
the fluid at any point of the circuit, and fds line integration once round the 
circuit. 
f References distinguished by the initials Y. M. are to the part already 
published of the author’s paper on Yortex Motion. ( Transactions of the 
Royal Society af Edinburgh , 1867-8 and 1868-9.) 
f The general limitation, for impulsive action, that the displacements 
effected during it are infinitely small, is not necessary in this case. Compare 
$ 5 (11), below. 
