718 Notes on the Method of Dimensions. 



The correct determination o£ this number will follow from an 

 adequate physical conception of the problem in hand, and if 

 we have formed such a conception, the precise manner of 

 embodying it in the initial equation is a matter of taste or 

 of convenience. 



14. Bemarks on disregarding small quantities. 



In giving the solution of the problem of the vibrating 

 spheroid Lord Rayleigh says : " The frequency of vibration 

 of a globe of liquid, vibrating in any of its modes under its 

 own gravitation, is independent of the diameter and directly 

 as the square root of the density." Surface tension and 

 compressibility are excluded from consideration by the terms 

 of the statement. Viscosity is disregarded, and it is thereby 

 tacitly stipulated either that the liquid shall be an ideal one 

 tree from viscosity, or that the circumstances shall be such 

 that the variations of viscosity which occur among real liquids 

 can have no appreciable influence on the frequency. 



It has been shown (section 8) that in the absence of this 

 proviso the solution is given by equation (25), which may 

 equally well be written in the form 



CT = ^^_Ji_\ = V^(II). . . (61) 



If juu = exactly, i|r(II) = i/r(0)=constant, no matter what 

 the form of \jr ; and if //, has no effect on a, yjr(lH) must be a 

 mere constant. In either case we have the simpler result 



cr^NvVy. ...... {62) 



But all real liquids are somewhat viscous, and it is interesting 

 to inquire what may be the effect of a small but finite 

 viscosity. 



For given D and p, as fi approaches zero, II approaches 

 zero and ^(11) approaches ^(0)=N. If ^(11) approaches 

 i/r(0) slowly ) ^(11) is sensibly equal to N for all small values 

 of //,, and equation (62) js sensibly accurate. But since 

 nothing is known, a priori, about the form of -^, it is always 

 possible that ^r(TT) may change very rapidly near 11 = 0, with 

 what amounts to a physical, though not a mathematical dis- 

 continuity. And if it does, the variations of even a very 

 small viscosity may influence the frequency. 



That such a form of the unknown function is not an im- 

 possibility is easily shown by returning to the air-resistance 

 equation given in section 13, namely, 



