MOTION IN RELATION TO THE SURFACE FRICTION OF FLUIDS. 
213 
fluids or in the same fluid. Then, if a, a 1 , ..., be homologous lines in the different 
systems, T, T 1 , ..., corresponding times, such as the times of oscillation, c, c l , ..., the 
maximum excursions of similarly situated points in the fluids and the equations 
dp _ \d 2 u d 2 u d 2 u\ du 
dx ^ 1 dx 2 dy 2 dz 2 j P dt 
&c., &c., 
(which are identical with the general equations of motion of a viscous fluid, in which 
the terms involving the squares of the velocities are neglected) are satisfied for one 
system, they will be satisfied for all the systems provided 
and 
fie., provided 
c 
u ™ 
u 00 v OO w, X 00 y CO z, 
uU pUX 
f) oo — oo -, 
X t 
x cc a, t co T, and 
a /x 
- OO —~ ■ 
T e 
From this last relation it follows that for dynamical similarity the value of 
ua/v must be constant for all the systems. 
In 1873 , Helmholtz,* in a paper to the Royal Prussian Academy of Sciences, Berlin, 
gave a somewhat more general treatment of the question. Considering two fluids of 
densities p u p 2 , and kinematical viscosities v 2 , the conditions under which the motions 
of the two fluids are similar are determined thus :—- 
Taking the equation of motion of the first fluid as 
1 dp _ du du du du _ \ d 2 u d/u d 2 u ] 
py dx dt dx 1 dy dz 1 [dx 2 dy 2 dz 2 j 
and two similar equations, and writing 
v 2 - ( Pu P2 = r Pi> 
then in order that these equations may be transformed into the equations for the 
second fluid 
ldP dU n dU v dU . w dU [d 2 U d 2 U d? U1 
P2 dX dT dX dY dZ l ' 2 1 dX 2 dY 2 dZ 2 l 
&c., under the given conditions of similarity 
U = nu, Y = nv, W = nw, 
* Helmholtz, ‘ Wissenschaftliche Abhandlungen,’ vol. I., p. 158. 
