446 Mr. R. Hargreaves on 



potentials is important. The positive character of T / = Sp"7~ 

 will appear later (cf. end of § 34). P 



§31. The above method proceeds directly to the energy- 

 integral without taking note of the values of FGH. The 

 solution ^ L attaches to the density p{ax + c'y + b'z), or with 



principal axes to the density pscja 2 , and therefore (%i<:r) ~ 



to i y /V = pw 3 ic/V ; i. e. for w^ G is (%i« 2 ) -^ , while F is 



— (%2^ 2 ) -rl • Hence for a general rotation, 



2 P F=2y 3 (</« 2 %1 -/ ^ 2 ) =2 ^ (>j» 3 -rco 2 ). (118) 



Any influence of the vector terms on the scalar potential 

 depends on 2pF, and therefore vanishes when the axis of 

 rotation is that of translation. 



The forces electric and magnetic due to the scalar potential 

 are odd functions of xyz (first order), those due to the vector 

 potential are even functions (zero and second order). We 

 shall consider the contributions which appear in the equations 

 of energy and of mechanical force, referring to (56), (58). 

 and (59), when integral values are taken. 



First take the term ^X'i x dn, to which the scalar element, 



with X' = |(V + ^ + M ':), contributes an effective 



element. The term connected with o> 3 is 



J2X'^= *£p ^T.f.flKo'.r + M y + L„'-- ) - y (Lo* + «»> + M,'*)] 



— -j^— :(A — i5)JSo — -jTT 3 ^ ' ° 



= — - M3 («— HpyJ o -y«i ■ • • ( 11J ) 



If all components are taken 



J 



2X'Wt«= ^p26> 3 (a 2 -& 2 )N '. 



When L 'M '!N" ' are compared the factors which are peculiar 

 to N ' are ^(c 2 + X), from which follows Xr'N , (a 2 -b 2 )=0. 

 Thus the above expression vanishes if q) 1 :cc 2 : co 3 =p : q:r, or 

 if the axis of rotation is also that of translation. 



The expression in (119) divided by © 3 is a couple about the 

 axis of z. As the components of a uniform translation taken 

 in the varying directions of a rotating ellinsoid are periodic 



