18 Mr. G. F. Fitzgerald on the Mechanical 



be multiplied in order to obtain the total energy. Calling the 

 quantities of energy Q x , Q y , Q 2 , we thus get 



Q x =m%nv% 



^ = M^Snw V!=7? sin cf>, 



Q 2 =Mktnv s ^T=f? cos cf>. 



In order to be able to perform these summations, it is ne- 

 cessary to know the mean values of nv, nv 2 , and nv d in terms 

 of fi and <£ ; and I shall, in the first place, merely assume that 

 they can be expanded in a series of spherical harmonics, thus : 



S v 



nv =-^r(A + A 1 + A 2 + ...)^^, 



^ = 5?(B' +B 1 +B 3 .+ ; ..)^#, 



4z7T 

 47T 



The effect of this is to obtain our former results under the fol- 

 lowing simplified forms. Our first series of equations gives 

 A x = ; and as A l must be of the form 



A 1 = a 1 fi + a 2 \/l —fj? sin + a 3 \/l —fju 2 cos <£, 



we get 



tt 1 = a 2 = <23=0. 



The second system of equations gives 



WNv 2 

 P**=^JJ(B + B 2 >^#, 



MNv 2 

 ? m = -^jJ(B + B 2 )(l-V) sin 2 <£<?«#, 



P -= ^T lf( B o + W "V) cos 2 4, dp dfc 



M~Nv 2 



F^= P*y = —J -JjB 2 (l — yLt 2 ) Sin (f) COS <£ dyU, c/</>, 



■P«b= P^ = —7 — J J B 3 ftv 1 — fi 2 cos d/jb d<f>, 



MM 7 . 



P^= P ya ,=— — JJ B 2 /Ltv 1 — //, 2 sin dfi d<p. 



