hersey: derivatives of physical quantities 627 



the size of the instrument. Let D, I, and denote, respectively, 

 the diameter and length of the tube and its angle of inclination 

 to the horizontal, d and p the diameter and density of the 

 ball, p and p the density and viscosity of the liquid, and t the 

 roll-time 7 in a locality 8 of gravity g. Assuming that a complete 

 relation does subsist among these quantities, the n-theorem 

 shows that any equation describing that relation, whether ob- 

 tained theoretically or experimentally, must be reducible to the 

 form 



M 



P D 



- 1 = f unct. fe g -^-, shape) (30) 



2 \p n 2 / 



the shape, in turn, being fixed by the arguments y^, jz, and 0. 



Taking t, g, and D respectively for Q , Q if and Q 2 gives a = 0, 

 /So = - 2, a = 1, and /3 = 3; so that by (18) 



tdg 3 3 t dD 



An interesting check on (31) is afforded by differentiating the 

 empirical equation for such an instrument. 9 The equation has 

 been presented in the form y = a + bx, in which x denotes 



-V*(?-i) 



and y denotes vl\D*g[- 1 ), r being the roll 



j • 



time per unit length — , v the kinematic viscosity — , and a and 



I p 



b particular numerical values fixed by a particular choice of — 



and 0. Recast in the form (30) it becomes 



7 That is, the time required for the ball to roll down. This instrument, pro- 

 posed by Flowers (Proc. Am. Soc. Test. Mat., 14: 565. 1914), is further discussed 

 by the writer in this Journal, 6: 527. 1916. 



8 Having set up such a viscosimeter in Cambridge, the question arose whether 

 there would be any sensible change upon taking it to Washington, where gravity 

 is 0.3 per cent less. The conclusion is that the roll-time in a very viscous liquid 

 will be 0.3 per cent greater in Washington; and that the effect of gravity dimin- 

 ishes when the fluidity of the liquid increases, falling to 0.2 per cent for water. 



9 This Journal, 6: 528, eq. (6). 1916. 



