438 Mr. R. Hargreaves on 



It is clear that S in comparison with S s is not so much 

 reduced as the factor l — %p 2 would imply, and is more reduced 

 than the factor (1 — Xp 2 )i would imply, i.e. S>S S (1 — Xp 2 ) 

 und <S S (1— -Xp 2 )* ; a conclusion which is confirmed by the 

 last form of J' in (104). Apply to (106) the inequalities 

 J'>J/(l— tp 2 ) and < J/, and it appears that 



T >&%p 2 /(l-%p 2 ), and < 2SX//(l-2p 2 ), j 



or since I (108) 



E = S + T, E>8/(1-Sp 2 ), and <S(l + ^ 2 )/(l-:%> 2 ) ; J 



and if we apply S>S S (1 — Xp 2 ) and < S s (l — Xp 2 )~ to these, 



T>S, ? V, and <2S,X/> 2 /Vr^Xp ) 



, ,1 f. (109) 



E>S S , and < S fi (l + X/)/x/l-X^ 2 j 



What orientation gives a minimum value to S for a given 

 value of 2w 2 or %p 2 ? 



The general stationary condition 



^s . us i is i ^s 



= ku..., or - — = - -y- = 

 aw jo dp q a<] r ar 



leads to a=6=c, which denotes the obvious indifference of 

 the sphere to direction. Take then a change from r along c 

 with no p or q, to p^r with jP 2 + Q 2 + ^ 2 = ?\m JP and ^ small. 

 Then 



J' = (a 2 + A) (7> 2 + X) (c 2 + X . 1 - r 2 ) 



- W-O (/> 2 +X)-X^(^ 2 -^ 2 ) (a 2 +X). 



•■• r^>-.r^[ i+ fl? {( '- -v(i ' + ' x) 



If c is the greatest axis, the value given by the use of J/ 

 with r , j) = q = is a minimum ; if c is the least axis it is a 

 maximum, and if c is the mean axis change depends on 

 direction. Since S is a momental expression for energy we 

 expect a minimum to correspond to stability, and that 

 requires the greatest axis to be in the direction of the 

 translation. 



