Hi/) - /(.^O = Y [/' (.'/) + f (*''')] - ^ I /'" ('" + '^) (^' - ^ ^') '^^^ 



( 108 ) 



As j ?/ ƒ" (/«) </?t = and /'" (m 4- 7/) — /'" {7n) — f/'"! {m + r) dv, 

 we may write instead of j u f' {m -\- u) du , the double integral 

 I ?^ du \ ƒ ^^^ (?n -|- v) do which by reversing the order of integration 

 is transformed into | ƒ "' (rn -|- v) dv j ?; fZ?«. 



-V, 



I now proceed to integrate with respect to u, and so we obtain 

 the following relation : 



The operations may be repeated and by doing so we shall find : 



fiy) - /(•'-•) = y lf{y) + fV)'\ - ^i if'' ill) + r' (•^•)] + 



The expansion may be easil}' continned in the indicated manner, 



but for the end I have in view that deduced above goes far enough. 



If according to this formula we replace i, — i^ hy 



r Tj" 



— (Sj + i'l)— — - (^3^'* -f ^1^^ ), terms of the fifth order are neglected, 



and if we replace (3 z^ .\ — 2 ^,"i) - (3 z, z, — 2 ^^"1) by 



^ (8.^^ + 3z,z, - 2.,^v -\- 3i,^ + 3.,ë, - 2e,iv), 



we neglect a quantity of the third order. I propose to terminate 

 the expansion of 2 with the term in Tj', then the above 



yp 



mentioned substitutions will not alter the order of approximation. 

 So we obtain the following approximate formula : 



