320 REPORT— 1873. 



f ^^ . ^^ \' 



Now if we introduce two new variables, (u) and (««'), and assume them to 

 satisfy the following two equations, 



we shall obtain, of course, 



dx_^^_ (1-M^)v/R^ ^^ ^ (1-^^)V^ 

 du li\{x^—«^i) ' f^«' A(^2-'^'i) 



when we remember that 



(1 -X\rO(l - ;u=^0-(l -X^^g(l -^^r, ) - 

 (X=-;x=)(.^',-.r,)=;u\Or,-.^'J. 



From these equations we are able to obtain 



c?M ' du' ' du ' du' 



in terms of a? .^^ ; also the ratios ' J^^,-^ — \ give us relations, from which we 

 are able to deduce the following expressions : — 



Jj^ ^^-^2 ^ 1!^ K 3 (^» ^) , 02. 3 (". ^<^) 

 V IT ^ ^ ^I'-^'a ^ ^l^fc 03. 2 (^> •^) 02. 2 (^. ^) 



^ ^^^/^ . c?V(l-a;,)(l-a:,) ^ _ _^ . 0. , 3 (^> •^) 0ua1^'J<O, 

 VAr^Xi/ij rfw 2/i^ ,^„^ ^ (t;, tt;) ^^^ „ (v, w)' 



^ ^V . ^^V(l-^i)(l— a-o) ^ ^ ^ 0;, 2 ( ^, w) 01 , , (v, lo) 

 V A:/,^j ■ du' 2/.,^ * 0„_ „ (v, iv) ^„_ „ (v, iv) 



Section 9. — Eosenhain deduces from the Table the following equation : — 

 i02, <ii»3. {0,, i'>^ + ^'> w + w') 0„^ „ (v -v', w-w') 

 -00, ('^ + ^'' ^ + «'') 01, (■^ -^'» ^-'^')} 



= 02, 3 (^> ^t') 03, 3 ('"> W) 00. 3 (^'' ^') 01, (^'. ^f') 

 -02, 2 (^'. W) 03, 2 0'» '^) 0,, 2 C'^'- "'') 00, 2 0'' '")• 



