§ 7 OF ARTS AND SCIENCES. 379 



whence, by differentiation with respect to Z', 



±=- ^-^' V J_ -U 3/ (4/-30+3(/-/') \ ^ 

 dT ^r — zl "^ T^^ T\ (4/' — 3/)^ )dT 



= — ^ + -^(^ ^^7_ \ ^ ' ^^^"^ ^•' ^^^'^'^^ substituting 



for its value, — '- (see hist section), we have 

 de e I e/ , eV 



^r r ' r(4/' — 3/) ' 4/' — 3/ 



T \^r-^l) ^ 4/'-3/' 

 hence again from I. 



de 4e'' I eV 46'' , e^ 



Now from I. we have 



4/'er-3/er=3/-3/'; 

 /'(4er+3) = /(36r+3); 



I 3+4^^' 



which, substituted in the precedmg equations, gives 



\.a. 



^^ 4, ,. I ^^ ^ ± £== 4- ^' (3 + 4^^) 



dl^ 3 I2 + I26r _ ,3 3 



3+4^^ ^ 



r= iie^ 4- e^ 4- 4. £3 7; so that, finall}', 

 3 3 



i^ = T ^^ + T^^^' II 



^Z 3 3 Ai. 



