194 Dr. C. Chree on Applications of Physics 



If the self-gravitation were negligible, the denominator in 

 (52) would become unity, the numerator remaining un- 

 changed. Thus self-gravitation reduces the change of form 

 produced by the disturbing forces depending on the harmonics 

 of degree i in the ratio 



gpa lbi(2i + l)m*-(8is + 6i*-2i-9)mn+(±i*-2i 2 -3i--3)n* 

 1 n ' b(2i + l){3m-n){{2i 2 + 4;i+ e d)m-{2i + l)?i} '^ 



If n/m=0, 



or the material be incompressible; (52) reduces to 



pa^Y/i(2i + l)/{2(i-l)(2i* + Ai + 3)n} 



l+{gpa/n)i/(2i* + 4i + Z) ' ' " **?> 



and the ratio (53) becomes 



1:1 + (gpa/n)i/ {2i* + M + 3) . . . . (55) 



When i=2, (52) becomes 



_ pa?Y2(5m — n)/{n(ldm — 5n) } 



tt2 ~~ l + typa/n)(10m 2 -5mn + n*)/{5{3m-n)(19m-5n)}' ^ > 



A result equivalent to (56), with the notation 

 \ = m — n, P>=n, 



was given by Prof. Karl Pearson, in Todhunter and Pearson's 

 'History,' vol. ii. part 2, p. 425. An obvious misprint of 

 4//, for 14/^ occurs, however, in the denominator of his 

 formula. 



§ 27. We may safely assume m—n positive, so the nume- 

 rator in (52) has clearly the same sign as V/ ; also for a 

 given value of Y/ it diminishes as i increases. Thus so long 

 as the denominator in (52) exceeds unity there is no risk 

 lest the relative smallness of the forces proceeding from any 

 higher harmonic may be compensated in any way. It is 

 obvious, however, that the coefficient of gpa/n in the deno- 

 minator can be made negative by taking i large enough, for 

 ordinary values of n/m. For instance, if n/m = l/2, — the 

 is neom?is of uniconstant isotropy, — the coefficient of gpa/n 

 minator as a wnen 2 exceeds 9, and with i infinite the deno- 

 through the valu^ e woul d vanish and change sign as n passed 

 Ifwetakeasbefof^ 100 - 



6 "immes wt. per sq. cm., 



