jEolotropic Potential. 4:0 1 



respectively. For the differentiation of u a we have 



A (T -j " = — } Ua(lmn) + C»r + Bn 2 — 2A'mn, 



and A„ , = — a't*a(/ira») + 2(B7»iH-C7n— Amn i -rA'l 2 ) ', 



and the identity of results depends on 



au a {lmn) = Cm 2 + Bn 2 — 2A'tw«4- (fo + wtc' + /?//)-, and 

 la'ua{lmn)== B'lm + C'&i— Amn—A'l 2 + (/c'+ w6 + wa') {lb' 4- ma' -f nc), 



In the latter shape of (145), the relation 



2(AL +2A'L ')=A a <£ 

 is at once verified, while XpL + 2p'L ' = 2 involves a formula 

 which, if the form given above for j^ is used, amounts to 



•2tt -* d i da ' 1 v d f 00 tip 



A a Y tfAjtuupi A« l tlAJ x/A(ya) 



But 



^AW=AM=^ and /. Sl» E AW = x -%.= -^ . 



Thus 



1-11 





v/A(0) VA A A, 



In fact the two relations (95) turn on the relations (86), 

 which were requisite in the original proof. 



When the motional case is compared with the statical one 

 the extra factor k/u v i or (1 — 1p 2 )/\/l — %p' 2 + (%J>Q 2 appears 

 in the forms for potential or for energy. The. radical is 

 a form which occurs with ray-cosines. 



The connexion with radiation-formulas is less clear than in 

 the reciprocal case. For a moving standpoint -or replaces p, 

 and X /V 2 — £w 2 + U r 2 replaces V in the energy-formula p 4 /Y z . 

 Now V' 2 — 2! //,--(- U r 2 = V 2 // ; ,. and we have for an ellipsoid 



, . . p*(l-2r/ 2 ) f da> u . , 



//„=]. The energy-expression - — — M- ,. It the 



r 107T J U a Uj 



special spheroid zip [xyz)-=- constant is used as base, then zipl 

 occurs here, but in the general case a factor i/ p is replaced 



