ON QUADEATUEES AND INTEEPOLATION. 347 



the constants of integration. These constants must not be forgotten, but 

 as tliey disappear when the origin of integration is suitably taken, they 

 need not be further discussed here. 



The same treatment which was applied to the investigation of Cotes's 

 formulae may also be applied to a double integral, only then it is necessary 

 to use a series with odd powers only, because the even powers disappear 

 for + limits. "Writing 



u = a^x + a^^x^ + a^x^ + 



and taking the integral between limits + n 



-\ U„ = c. + 2^ a,,^3 + 4^5 «3^^^ + 5-^«3-^ + 



Assuming the first integral to vanish with x, c vanishes, and the first 



ditfi. Now writing in succes 



± 1. ± 3, ± 5 for a;. 



significant term is — aitfi. Now writing in succession 



1^ 



2 (— M-i + Ml) = «i + as + «o + 



2" (— «-3 + ^a) = 3ai + 3%3 + o^a-^ + 



2" (— «-5 + %) = 5a2 + 5^*3 + S^^o + . . . 



&c. 



or, using indeterminate multipliers 



1 3 



that is to say. 



Xi + 3\2 + 0X3 + ttX„ = ^— g n- 



Xi + 33X2 + 53X3 + nH„ = 475"' 



X, + 35X2 + 55X3 + n^K = g— ^n7 &c. 



stopping at w = 1, X, = -r 



, . ^ ^ , 567 , 51 



stoppmg at TO = 3, Xi = j-t^, X, = ^^ 



ff" 



u d:e- between limits -f- h is 



ff^ 



-g/i^ {—u_■^ + u{) 



tt dx^ between limits -I- 37j is 



mJ./ 



3 



^Q h?' (- 17«_3 - 189«_, + 189hi + I7W3) 



Similar formulae, symmetrical to an ordinate instead of to an interval, 



would be obtained by writing 0, + 1, -f^ 2 for x. But the integral 



vanishes for a: = 0, and these formulse would evidently be less advan- 

 tageous than those of the odd series, for the same reason that in a simple 

 integral the even series is the better. 



