DISCONTINUOUS FUNCTIONS. 241 



Y to represent any ordinate, parallel to the axis 

 of J/, of the system of lines above described. 

 Then we shall have 



Y = * (a, b,c :r) (1) 



Where <i> (a, h, c r) must be such a func- 

 tion of (a, h, c iv) as will satisfy the following 



conditions. 



Y = i{x) ; when x < a, and greater than o 



Y = f, (x) ; when x <,h, and greater than a 



Y = {^{x) ; when x < c, and greater than b 



&c. &c. &c. 



Y = f„., (.r) when x < /?, and greater than /3, 



The means by which these conditions have been, 

 hitherto, fulfilled are, either by the aid of definite 

 integrals, or by infinite series. See Dr. Peacock's 

 report on certain branches of analysis, British 

 Association, 1833, page 250, where he has given 

 a review of what has been done in this important 

 branch of analysis. 



Required to determine the equation of a system 

 of lines containing one break. 



(8). Adopting the same notation as in Art. (7) 

 we shall have 

 1 i 



