Definite Integrals, ivith Physical Applications. 385 



Hence, if L be the least of the quantities a, a + 1, « + 2 &c.l 



a-1, a-2 &C.j 



then y = f(L) is the required equation. 



Now L is evidently = the least root of the equation sin ir (x — «) = 0. 



Hence as in Art. (18) we have 

 y = coefficient of- in — fix) h. 1. '- — — where /"(#)= , ; 



J X * ' X " % ' dx 



or if we put 



sin ir (x — 



IT COS 



^>-* = <M*) then y=/(O)-/'(O).0(O) + ( / ' ( ^-y o): i&c. 



* Fourier has expressed discontinuous functions by means of 

 Definite Integrals ; and when it is unnecessary that their form 

 should be explicit, his results may be conveniently used. 



* Vid. Fourier, Theorie de la Chaleur. 



