ox QUADRATURES AND INTERPOLATION. 353 



expansion, both odd and even powers of \/Z j but that E" + E~^ may be 

 expanded in integral powers of Z, and E"^ — E""' in odd powers of VZ. 

 Then the differential coefBcient of E' + E~' with regard to Z is 



X (E-' =F E— >) '^, in which ^| = 1 - E'^ = —-/whence 



and also 



^(E^ + E-) = |^(E- + E-) 

 E^ = i (E^ ± E-^) + i (E"= q: E-^) 



If the upper sign be taken, writing .... 



'k (E^ + E-^) = a + /3Z + yZ^-t- oZ3 + 



''^ .... .... ... 



suitably determining the coeflBcients a /3 y and giving the proper 



interpretation to M, furnishes the formula for interpolation symmetrical 

 to an ordinate ; while, if the lower sign be taken, writing 



1 (E- - E--) =ai VZ + agZ^ + ag Z^ + 



furnishes the formula for interpolation symmetrical to an interval. The 

 coefficients may be determined either by the ordinary methods of in- 

 determinate coefficients, by the calculus of generating functions, or by 

 ■writing 



«x = c_„ «_„ + + Cot'o + Ci?«, + C„M„ 



and determining the coefficient c in the simplest form, so that x ^=rh shall 

 make c^ = 1 and all the rest vanish.* 



The actual formnlse are best expressed in a notation similar to that 

 originally given by Newton and Stirling, namely, for th^ case symmetrical 

 to an ordinate, in which the differences run, 



^u_i A3i(_2 A^tt.g 



ll-Q A%_i A-'it.j A^teg ...... 



Am, A3i{_, A^iLa 



write B = - (Am_i + ^U\) 

 • • • C =-| (A%.2 + A%_r) 



■ D = '|- (A5tt_3 + ^H_^) &c. 



and a = u^, h = A%_,, c = A* . u_2 &c. 

 so that C = BZ, D = CZ = BZ^ 

 and b = aZ,c = VL = a7? &c. Then 



* See a paper by the author in the Messenger of Matlwmatics, vol. ir. p. 110. 

 Another proof is given by Professor Emorj- McClintock, of Milwaukee, in the 

 Amei-ican Journal of Mathematics, pure and ai)plicd, vol. ii. 



1880. A A 



