113 



I, is an approximate value of e, and we have the following relation 

 between e, and e, , 



which gives, by the theorems before referred to, 



/(O being a function of e, and /, -^/^ being formed from /, ■»//, by 

 changing « to s^ . We have also, by (S') (T') (f/')j 



and therefore, by (X'), 



/x 2 »-^ 



in which, 



y^o ^' '^'' = ~ ^(fl « ^'+1 • r+2 ' '' = (7T"^7 • 



The development (^') is one form of the integral of the partial 

 differential equation (PO ; another form of the same integral, may 

 be obtained from the expression (XO for the characteristic function 



of a plane system, by changing a^ to j^rj » ^^^ ^y ^^ ^' ^"^^ ^^> 

 VOL. XVI. r 



