Motion of Solid and Fluid Bodies. 155 
In perfect fluids the terms arising from v, may be neglected in comparison 
with those produced by v,, at least in hydrostatics; though I think it probable 
that in hydrodynamics the use of the function v, + v, would be sometimes neces- 
sary. Also in viscous fluids v,, and the terms produced by it, will be very small, 
so that in this case we should always be compelled to use the function v, + v,. 
I now proceed to the investigation of the equations of equilibrium and motion 
of fluids, or bodies in which y= y,. The general equation of equilibrium and 
motion in this case will be 
GSS (x8E + ven + 28¢) dm = SS iv, dadydz, (5) 
we ( \ "Fup -p'sinddpdodg. 
The value of p’, in terms of &, y, ¢, &c., to be substituted in this formula, is thus 
found. Let 2, y, z, wa, y+, z+, be the coordinates of the positions of 
rest of m and m’; then in the altered position of the molecules, if x, y, z become 
vu+t& y+, z+, the coordinates of m’ will become 
1 
rpétat Fat + 7 b+ho 
where 
og Pape ‘al 
ot gt Bie a wh te 
2 oie +7048 
and consequently p-+ p’, being equal to the square root of the sum of squares of 
differences of coordinates of its extreme points, will be expressed by the formula, 
tra [(otdbenred)s (+ bead eit'+ (bor SS} 
from which, by neglecting the smaller i Aa and assuming 
_ dy i dg dy 
~ dz ag an da +5 a Etk dx? 
we obtain 
ptp =pt+ Az nes = cos*y--ucosBcosy-+-v cosacosy--w cosacosp 
and, finally, 
