188 Mr DE morgan, ON INTRODUCING DISCONTINUOUS CONSTANTS 

 proceed alternately from the curve to OA, parallel to the axis of x. 



and of OA to the curve parallel to the axis of y. The ordinates of 

 the successive points of the curve P,, P^, P,, &c. are the values of 

 (px, <p'^x, <l>'x, &c. for the given value of x. 



The general expression <ff-x, is then one which requires a develop- 

 ment of the following kind, 



.r, C„_,, + .T„ Cj_, + X3 Ci^„ + ... 



where x^, x^, x^, &c. are specific quantities depending upon the func- 

 tion in question, and C„ ,, means 1 when x lies between a and b, and 

 for all other values, &c. 



Let (px = fix + yx .<paX (1), 



where ax, fix, yx, are given functions, not periodic, and </>.» is to be 

 found. In my treatise on the Calculus of Functions already alhujed 

 to, it is shewn that the complete solution of the preceding is 



^x = nx + vx . ^ax, 



where liX is any particular solution, vx any particular solutidn of 

 (px = yx (pax, and ^ax the general solution of (px = (pax. 



Among the solutions of (1) is the series 



fix -f yX . fiaX + yx . yaX . fia?X + yX . yaX .ya'x fici'x + 



obtained as follows : 



