134 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
p; u, ¥, w, Shali be continuous functions of space and time, and it can be shown that 
this involves certain conditions between the distribution of the mean-motion and the 
dimensions of Sp and 7. 
Mean- and Relative-Motions of Matter. 
Whatever the motions of matter within a fixed space S may be at any instant, if 
the component velocities at a point are expressed by u, v, w, the mean component 
velocities taken over S will be expressed by 
— (pu) 

> 
Fe Cet RM MGA tities. (ah) 
If then u,v, w, are taken at each instant as the velocities of x, y, z, the instantaneous 
centre of gravity of the matter within 8, the component momentum at the centre of 
gravity may be put 
pu pu + pun. ve Yo. anitiiy cake OS 
where w’ is the motion of the matter, relative to axes moving with the mean velocity, 
at the centre of gravity of the matter within 8. Since a space S§ of definite size and 
shape may be taken about any point a, y, z in an indefinitely larger space, so that 
x, y, 21s the centre of gravity of the matter within 8, the motion in the larger space 
may be divided into two distinct systems of motion, of which wu, v, w represent a 
mean-motion at each point and w’, v’, w’ a motion at the same point relative to the 
mean-motion at the point, 
If, however, w, v, w are to represent the real mean-motion, it is necessary that 
= (pu’), = (pu’), = (po’) summed over the space 8, taken about any point, shall be 
severally zero; and in order that this may be so, certain conditions must be fulfilled. 
For taking «, y, 2 for G the centre of gravity of the matter within S and 2”, 7, 2’ 
for any other point within §, and putting a, b, ¢ for the dimensions of S in 
directions «, y, 7, measured from the point 2, y,z, since w, v, w are continuous functions 
of x, y, 2, by shifting 8 so that the centre of gravity of the matter within it is at 
wv, y’, v, the value of u for this point is given by 
au 
(v’ — x)? + &e. (6) 
da? 
wl— 
isi, +@ —o(F) + —y() +6 -9(Z) + 
dy} g /9 
where all the differential coefficients on the left refer to the point «, y, z; and in the 
same way for v and w. 
Subtracting the value of w thus obtained for the point a’, y’, 2’ from that of w at the 
