94 Mr. Jarrett on Algebraic Notation. 



2m+2t^-2 r— 1 I I 



00 em+2r— 2 i ^ 



+ COS mx . S,. \ n .\ m + 2r-2 . ^,n+2r-z im + 2r-2 -if:^ I ' 



m+2r— 2 r— 1 I I_ I 



by the converse of Art. 12. 



Ex. 9. Laplace's Theorem : If 



7/ = yj^ {z + x.(j){i/)}, 



wlieii z is independent of .r and 3/, then shall 



°? x' 



For, by Maclaurin's theorem. 



Put z + X . (p {y) = u, then y — ^ (m), 



and ^ = ^' • '^ ^"^ = ^■'U.d„.^\f{u) ^dji ^ d^ (z + x . (p {y)) 

 d.y d..^{u) d.u.d„.^{u) d.u d- {z + x . (p iy)) 



^ <p( y) + x.d,.<p ( y) ^ <t> iy) +x.d,y.d,.<p {y) 

 \+x.d,.^{y) I +x.d=y.d^.<p{y) 



Whence, mnltiplying' up, 



d,y {I +xd..y.d,.<p{y)\ = d-.y {<(> (y) + x .d,y .d,.<p{y)\; 



and, cancelling identical terms, d^y = d^y . (p^{y). 



Again, d, .f[y) = d,y . d,f{y) 



= d:y.(f){y).d,f{y). 



