Sir W. R. Hamilton on certain discontiimous Integrals. 293 



the coefficient of ocn in the development of (1 +x'^)~^; there- 

 fore, 



P2,_, = 0, whenp = 0, {55.) 



and, in the notation of factorials, 



P2.= [0]-^[-i]'=(-iy7r-y;'^rf^sin^^'-; {56.) 

 so that, by (54.), 



G (0) = - (P2,+ P2.+2+ ... ). . . . • (57.) 

 when p = 0, and when n is either 2 i or 2 i—1, 



8. For the critical value p = \/ 3—1, we have, by (38.), 



r(p) = ^2(„foP„; (58.) 



therefore, for the same value of p, by (46.), 



G(p) = i2;;^ip«-P(„f.p« 

 = i(Po + Pi+ ..• + P„-i-P,-Pn+i-..-); (59.) 



so that the discontinuous function G, like F, acquires, for the 

 critical value of p, a value which is the semisum of those 

 which it receives immediately before and afterwards. 



9. We have seen that the sum of these two discontinuous 

 integrals, F and G, is always equal to the sum of the first n 

 terms of the series (12.), so that 



F(p) +0{p) = Po + P, + ... + P„_,; (60.) 



and it may not be irrelevant to remark that this sum may be 

 developed under this other form : 



in which the factorial expression [n]'' [0]~'' denotes tiie co- 

 efficient of .r* in the development of (1 +x)"; and 



Q* = 2l-/3^^^(3-l)' (62.) 



Thus Pq = Qo ; ■>! 



Po+P. =2Qo+Q,; I 



P„+P, + P, = 3Qo + 3Q. + Qo;r ■ '^ ^ 

 &c.; J 



and consequently 



Po = Qo ; -1 



P. = Qo + 2Q. + 0,; f ^"^^-^ 



&c. ; J 



which last expressions, indeed, follow immediately from the 

 formula (10.) 



2 7r»/ -TT 



