28 Prof. J. J. Thomson. 



Substituting for Sa and SI, their values, we get 



2 fa dA 2 dA 2 V c?a? 2 ^ cfo<% / 



Similarly, 



If we neglect the squares of hx t &y, Bz f Sw, <ty, Sg, 



dx'dy , dzdu}'d^'dy'd^=dxdy,dzdwd^dyd^ ^ 1 + _|_ + ^-^ T 



du> c££ J 



Now ^+^+^=0. 



cZa? efo/ 



And by the equations written above 



d.St»_ L d*Q 



and d^-d^n <m = 



dg d v d£ 



dy* + dz* J J ' 



-p , d 3 Q d°Q d*Q n 



But ^ + ^ 2+ -^ = ' 



and since 



<PV_ f I * V_ d £ d_ 1 2Q 



<2 2 Q . . ' •' 



we see that — - is a homogeneous function of iy, £ of zero dimen- 



dh 2 



sions, and therefore by Euler's theorem 



{ r d d d \ d*Q rt 



