626 hersey: derivatives of physical quantities 



Q?Q2° ■■■Q K k- Qo=funct. (Q? Qf.--Q£- Q k+1 , and other n's) (26) 



The values of a, (3, etc, can now be read off directly by identi- 

 fying them with the corresponding numerical exponents in the 

 equation, of type (26), afforded by the particular example in 

 hand. 



I. In the case of a journal bearing, under certain restrictions, 

 we may expect a relation of type (1) to connect the coefficient 

 of friction /, with the viscosity of the lubricant n, the revolutions 

 per unit time n, the bearing pressure p, the journal diameter D, 

 and the volume of oil V forced through the bearing in unit time. 

 Let it be required to calculate the effect of altering the size of 

 the machine from a test in which nothing is varied but the rate 

 of pumping oil through the bearing. By the Il-theorem, 



/ = funct. (®^, ^, shape) (27) 



\ V p / 



Let*/, D and V serve respectively as Q , Qi, and Q?.. Compar- 

 ing (27) with (26), «o = 0, /So = 0, a = 3, = - 1; hence, by (16) 



and (17), a = and b = — 3 — , or 



D 



y = -ziy ' (28) 



dD DdV K 



Also, by (20) and (22), A = 0, B = 12 —^ and C = 9 (-Y; there- 

 fore 



w.uZ^Wnvay (29) 



Equations (28) and (29) enable us to predict the bearing losses 

 of any slightly larger or smaller machine in the same geometri- 

 cally similar series. This requirement of geometrical similarity is 

 an instance of the constant-product restriction. The products 

 in this case are the length ratios fixing the shape. 



II. Let it be required to find the effect of gravity on a rolling 

 ball viscosimeter in terms of the effect produced by changing 



