250 THIRD REPORT — 1833. 



and if we make, therefore, 



y=^'D."-^«^„+^W-i^*^(»'.+^') (3.) 



the equation will be true for the ordinate of every point of the 

 sides A C and C B of the triangle ABC. 



More generally, if we suppose y = f^ x, y ■=. (p^x,y ■=■ (^^x, 

 y = (p^x, &c., to be the equations of a series of curves, then the 

 equation of a polylateral curve composed of the several portions 

 of the separate curves corresponding to values of x, included 

 between the limits a and b, b and e, c and d, &c., would be, 



+ ('D/-f-^,)fcx + &c.; (4.) 



the value of the ordinate at each successive limit being replaced 

 by that of the succeeding curve. In this manner, if we should 

 grant the existence of the sign of discontinuity, we should be 

 enabled to represent the equations of polygons, and of poly- 

 lateral curves of every description. 



It remains to consider the nature of the expressions which 

 are competent to express ^Dj". 



The expressions which have been generally proposed for this 

 purpose are either infinite series, or their equivalent definite 

 integrals. Le Comte de Libri, however, a Florentine analyst 

 of distinguished genius, has proposed* a finite exponential ex- 

 pression which will answer this purpose. The examination of 

 the expression 



would readily show that its value is 1 when x is greater than 

 a, and that it is when x is equal to or less than a. It will 

 therefore follow that the product 



^(logO) e^^^s")'^- ") ^ ^(logO)e^'^e*'>(*-^) 



is equal to 1 between the limits a and b, and is equal to at 

 those limits, and for all other values. And, in as much as 



* Memoires de Mathematique et de Physique, p. 44. Florence 1829. The 

 author has since been naturalized in France, and has been chosen to succeed 

 Legendre as a member of the Institute : he has made most important additions 

 to the mathematical theory of numbers. 



