710 Motion of Solid Bodies through Viscous Liquid. 



be admitted, the general law of dynamical similarity requires 

 that for the whole resistance 



F = cpv*Z6*V* (48) 



where I is the length, b the width of the blade, and c a 

 constant. Mr. Lanchester gives this in the form 



F/p=cv*A*V*, (49) 



where A is the area of the lamina, agreeing with (48) if 

 I and b maintain a constant ratio. 



The difficulty in the way of accepting the above argument 

 as rigorous is that complete similarity cannot be secured so 

 long as b is constant as has been supposed. If, as is necessary 

 to this end, we take b proportional to n, it is b\/n, or V 

 (and not V//i), which varies as ?zV 2 , or 6V 2 . The conclusion 

 is then simply that bY must be constant (y being given). 

 This is merely the usual condition of dynamical similarity, 

 and no conclusion as to the law of velocity follows. 



But a closer consideration will show, I think, that there is 

 a substantial foundation for the idea at the basis of Lan- 

 chester's argument. If we suppose that the viscosity is so 

 small that the layer of fluid affected by the passage of the 

 blade is very small compared with the width (b) of the 

 latter, it will appear that the communication of motion at 

 any stage takes place much as if the blade formed part of an 

 infinite plane moving as a whole. V\ T e know that if such a 

 plane starts from rest with a velocity V afterwards uniformly 

 maintained, the force acting upon it at time t is per unit of 

 area, see (12), 



p\W{vlnrt) (50) 



The supposition now to be made is that we may apply this 

 formula to the element of width dy, taking t equal to y\ V, 

 where y is the distance of the element from the leading edge. 

 Thus 



Y = lp{v^fY^y-^dy = ^p{vlir)-Yn\ . (51) 



which agrees with (48) if we take in the latter c = 2/v'7r. 



The formula (51) would seem to be justified when v is 

 small enough, as representing a possible state of things ; 

 and, as will be seen, it affords an absolutely definite value 

 for the resistance. There is no difficulty in extending it 

 under similar restrictions to a lamina of any shape. If b> 



