214 
MESSRS. T. E. STANTON AND J. R. PANNELL ON SIMILARITY OF 
it will be seen that, multiplying (l) by — the required transformation is effected if the 
change in the scales of length, time and pressure are 
X=^x, 
n 
P — n 2 rp. 
It follows, therefore, from the linear scale relation that, if l and L are corresponding 
linear dimensions of two pipes in which the fluids of densities p 1} p 2 , and kinematical 
viscosities v u v 2 are flowing, in order that the two motions may be similar 
L — or ——- must have the same value for each. Again, from the pressure scale 
I' x V V 
relation it follows that for similar motion the value of P /pV 2 is the same for 
each fluid. 
The method of Lord Rayleigh, which was first applied in considering the size of 
drops formed under various conditions,* is as follows. Assuming that the resistance 
depends solely on the velocity, linear dimensions, viscosity and density, and also that 
the resistance F varies as p'U'v 0 // then if M, L, T are the units of mass, length, and 
time, the dimensions of F are 
so that 
and therefore 
My T b / l y / m y _ ml 
xLy \T/ \LT/ T 2 ’ 
a = 1— d, b = 2 — d, c = 2 —cl, 
F oo p v 2 L 2 
) 
and the resistance per unit area can be written 
R = ^/(oL/a 
With reference to Newton’s Theorem on Similar Motions, in Proposition 32, 
Book II., of the ‘ Principia,’ the authors are indebted to Dr. R. T. Glazebrook for 
the following note :—- 
In this Theorem Newton shows that two systems of particles, if started similarly, 
will continue to move in a similar manner if the acceleration of each system is 
proportional to V'/L, V being the velocity and L a linear quantity defining the 
dimensions of the system—the diameter of a particle. Now denoting by f the 
acceleration and by p the density, we have for fluid friction 
LV' 
L 2 
= force per unit area = 
* ‘Phil. Mag.,’ 1899, vol. 48, p. 321. 
