382 PROCEEDINGS OF THE AMERICAN ACADEMY § 7 



Hence, substituting, we have 



d' 



dt^ 



=.6d' 



t — etc., <?r a — -— — — 

 dt' 6 dt^ 



e t d^e , ^ 



— -7=- —. — + etc. 

 6 dt^ ^ 



d' 



dt' 



=:2C' + t 



d'e 

 dp 



,3 d'e , t' d'e , , 



t —r- + 1— -\- etc. 



dt^ ' 2 c//^ ' 



= 2C' -\- 



tdM 

 dt^ 



2 



d'e 

 dt' 



-\- etc., 



whence c' ^ —- 



2 df 



t d^e 



— — -~ — etc. 

 2 dp 



de^ 

 dt 



= b' + t 



d'e 

 dp 



d^ , t^ d^e^ I P d^ 

 ~dW "^ 2~ ~~dP "^ T ~dF 



whence 



b' = 



d^ 



dt 



etc., 



P d^e 



dP 



dP 



6 dp 



-\- etc., 



and finally 



+ 



P d^e , P d^e , , 

 — — — . + — + etc., 



6 dP ~ 2J^ dP ~ 



from which the value of (f' can be determined. 



P d^€_ 

 6 'dF 



d'e 

 ~dP~ 



