the Method of Dimensions. 705 



omit it from the initial equation. It the constancy of the 

 quantity is an essential characteristic of the phenomenon, 

 we cannot expect to get a correct result by ignoring this 

 fact but must take account of it in some way. To put it a 

 little differently : the phenomenon which occurs subject 

 to the condition that a particular quantity shall remain 

 constant, may be regarded as a special case of a more 

 complete phenomenon in which this quantity is variable. 

 And it' a generally correct result is desired, the initial 

 equation must include all the dimensional constants which 

 would change the course of affairs if they did vary instead 

 of remaining at fixed values. 



The point to be grasped is that all the known physical 

 facts which are really pertinent to the problem in hand 

 must be utilized in order to get the best result possible, 

 and if any such fact is ignored the result obtained may 

 conflict with it. The easiest way to recognize the essential 

 dimensional constants of any problem is to include them 

 in the initial equation and treat them like any other 

 quantity, although there is also another possible method 

 of procedure which will be noticed in Section 11. 



8. Universal Constants ; Gravitational Oscillations 

 of a Liquid Spheroid. 



The manner in which universal constants are to be 

 treated is well illustrated by the problem of the gravi- 

 tational oscillations of a liquid spheroid, of which the 

 solution for a non-viscous liquid was given by Lord 

 Eayleigh in 'Nature' for March 18, 1915. For variety, 

 we may suppose that the liquid is viscous. 



Let a liquid spheroid of mean diameter D, density p, and 

 viscosity //,, execute oscillations of figure under its own 

 gravitational forces alone, and let a denote the frequency 

 of any one of its modes of vibration, — the slowest, for 

 example. If the small heating effects of dissipation be 

 ignored, a can depend only on I), p, yit, and the gravitation 

 constant y ; and the initial equation is 



F(<r,D, p,ix,y) = (21) 



The gravitation constant being defined by the equation 



■ f=y~, ....... (22) 



its dimensions are 



[ 7 ]= [,„-^r=] (23) 



