348 REPORT— 1880. 



These formulfe are, however, rather cui'ions than useful. For the 

 treatment of multiple integrals, symmetrical differences are more con- 

 venient. 



Section 5. — Quadrature by differential coefficients. 

 The reciprocal of the formula 



A =• £ dx — 1 



enables the difference between a definite integral and the term of a series of 

 ordinates to be expressed by means of the successive differential coefficients 

 for values corresponding to the extreme values of the fauction. Calling 

 these Uy and u,, 



^"' Unh - A-' Mo = I « *'^ - 1 I I "« - "o I > 



There is nothing indeterminate about this equation, the left-hand side of 

 which is the sum 



«0 -H Ml + ?/2 + + «n-l 



while the right-hand side has for its fii'st term^ / ^tclv, for its second 



term — (m„ — Uq) and for its general term thereafter, 



1.2.3 2r + 2^"-+''' \d^^^"'^~ 5^^"°/ 



where Bor+i represents Bernoulli's numbers taken without regard to 

 sign. 



The complete formula in its usual form is 



^ I y «*o + "i + «2 + + "„-i + "2" '«« j 



=J''\dx + I |^(m'„ - m'o) - Jo f («'"« - «"'o) 



The meaning to be attached to — - and ^3 is that they are the values 



ax'' ax'' 



n iV'u , . , 



01 -j-^ when x is made severally equal to nh and to zero.* 



The use of this formula presents no difficulty except in one remark- 

 able case, pointed out by Legendre,t in which all the odd differential co- 

 efficients after some particular value of r take the same value at both 

 limits. Among these may be instanced u= n/ (1 — Jc"^ sin ^x), the 



limits being and — tt or tt. All the odd differential coefficients are 



affected with the factor sin x cos x, which vanishes at both limits, so 

 that each term of the expansion contains zero as a factor. Nevertheless, 



• See Woolhouse On Tntcrjwlation, Summation, ^'o., part ii. p. 45 (note), for a very 

 singular extension of this formula. 



f See his Fonctions EllijJtiques, vol. ii. p. 57 



