DISCONTINUOUS FUNCTIONS. '253 



Since A, B, C, &c. &c. are functions of .r, we 

 differentiate under this hypothesis. 



From Art. 10. we have A = 1 + "l^I^ .-. d A 



= d. 



d <n X 



= — (a in x) dx -{- (a — x) 1 , dx 



(a w xf 



dA (a <n z) av> X n • / m 



. . -7— = — 7 \ + 7 \2 = . since (o — xy = 



dx {a — x) ^ {a <n xf ^ ' 



{a m 5-)'. 



In a similar way we have -5— = and -7- = &c. 

 Consequently we shall have 



d Y„ ABC.t . A.-BC...t B.C ... t . 



-^ = 2^^ -t W+— ^^i ».W+ 2"-' 



Hence, equation (1) becomes 



(^-^)=^(«--) « 



(15). We shall now proceed to illustrate the 

 above general formulae, by the application of them 

 to a few examples. 



