92 Mr. J. H. Jeans on the Partition of 



The system is accordingly specified by 6N + m Lagrangian 

 coordinates, and the energy is some function o£ these 

 6N + 7W coordinates. 



§ 2. If there is no interaction between matter and sether, 

 this energy can readily be expressed as a sum of squares of 

 these coordinates. The energy is o£ the form 



\m t (<+«„» + tO +/ (£,, h, ■ ■ ■ ?»)• 

 The last term represents the energy in free aether which, 

 if we assume the exact linearity of the electromagnetic 

 equations for all electric intensities with which we shall 

 ultimately be concerned, is known to be 



The energy is not yet expressed as the required sum of 

 squares, for the quantities X, Y, Z, a, /3, 7, are not inde- 

 pendent Lagrangian coordinates. Let each of the quantities 

 X, Y, Z, a, /3, 7, regarded as a function of x } y, z, be 

 expressed in the form 



f {x, y,*) = — 8 J j / (A,^ 7 v) cos p (X-x) cos g (ji-y) cos r (y-z) 



- " dp dq dr dX d/u dv, 



and let the right-hand integral be further transformed into 

 the sum of an infinite number of terms of the form 



(X 2 + Y 2 + Z 2 + o? + /3 2 + 7 2 ) dx dy dz. 



CZipx + W + rz), 

 ain 



wh^re /, m, n are direction cosines, k 2 = p 2 + q 2 + r 2 , and C is 

 independent of x, y, and z. The value of any one of the six 

 quantities X, Y, Z, a, ft, 7, say the first, can now be expressed 

 in tjhe form 



-]-CO T 277 



X=A J I I (Xicos + X 2 sin)(# sin cos (j>+y sin sin $ + £ cos 6) 

 - 0000 dicsmOdOd^ . . . (1)* 



where X x and X 2 are functions of k } 0, and <£ only. 



The condition <— + ^— + *■=- =0 is now satisfied if 

 O.y oy ds 



Xx sin cos + Y : sin sin <j> + Zj cos (/> = ; . (2) 

 * |c/*. Whittaker's solution of y 2 V+V = 0, 'Modern Analysis/ p. 821. 



