330 Prof. L. Boltzmann on the Assumptions necessary 



are introduced instead of x',y',z\ so that the expression (38) 

 becomes 



dpdq'dt'd&'drh'd&'vdNdEi. . . . (39) 



Lastly, we introduce, instead of the spherical polar coordi- 

 nates N, E of the point H, its spherical polar coordinates S, yfr'; 

 so that we obtain 



vdN.dE=(rd&.dy'. 



Lastly, expression (30) becomes therefore 



dg dri d% d& d Vl ' d& ad$ . difr', 



by which equation (33) is proved. 



If we prefer to prove equation (37) analytically, fig. 3 

 would give 



x=rm, y=rjjb sin 0, z=rfi cos 0, 



where OR 1 =OE 1 '=^ 



8 : <f> = fJ> : cr, 8 : <£' = /-<<' *• o", fi'<j)'=fi^> 9 



s=mn + fivf, a- 2 d 2 = a 2 —fi 2 (j> 2 



= 1 — (mn + fxvf) 2 — /m 2 ^ 2 = (mv—fjunf) 2 , 



m! = ns + vad — mn 2 + jJ<v 2 f, 

 where 



n 2 = cos 2N, i> 2 = sin 2N. 

 From 



s = m!n + /j/vf = mn + ^1/, 

 it follows that 



fj!f=zmv 2 — fifn 2 . 



If in this equation and in the equation /jJ<j>'=fi<f> we put 



f — e cos ! + e sin 0', <£' = e sin 0' — e cos 0', 



it follows that 



/J cos 0' = mev 2 — fiefn 2 — /^e<£, 



ft' sin 0' = mev 2 —(jb€fn 2 + ^0. 

 By multiplying by r and observing that 



rm! = x\ rfju sin0'=y', rfju' cos 0' — z', 

 f=ecos0 + esm0, <j)=ecos0— £sin#, 



r/*/= e?/ + az, rfjbcj) =—ey + ez, 

 we obtain 



w' = w 2 # + j/ 2 e?/ + v 2 ez, 



y' = v 2 e# — (e 2 + e 2 ^ 2 )y 4- 2v 2 eez, 



z' = y 2 <?<2? + 2v 2 eey — (e 2 + A 2 )e, 



