Mr, Jarrett on Algebraic Notation. 95 



d,'-f{y)=dAd-.y-1>{y).d^.f(y)} 

 = d. {d.,.y.^{y).d,^.f{y)\ 



= d...{d.y.fQ/)\\d,./iy)}, 

 d\. .fiy) = <■ . d. \d:y . ^W • d, -fiy) 



= d..d,.{d.y.f(fi\\d„.f{,i/)\ 



= d\.{d,.y.^{^\d,.f{y)\ 



= d\.{d..y.'f(yj\\d,.f{y)\, 

 and similarly, d-,. .f{y) = d."'-' . {d.y . 0l^'" . d,.f{7/)\ 



= d='"-' {<p{y)'"-d..fiy)}. 

 Put X = 0, and y becomes x/^ (z) ; 



■• d\„ ./{y) = dJ"-^ . [H^"" ■ d- ./^^ {z)\. 



00 ^m 



and f{y) =fi' (z) + S,„ - . rf.™-' . {^Tm'" ■ d-. ■f^' (^)}. 



m 



Cor. Put -^ (u) = u, and we shall get 



J\y) =/(2) + -s-,,,^ .rf="'- . {^W ■ d~..j\%)\, 



which is Lagrange's theorem. 



Ex. 10. If X, X , z, and z' are independent of each other, 

 while y and y' are determined by the equations 



y = ^{z+x.(l>{y)\, 



and y= x/.'{s' +x'.(p'{y)}; 



