244 NEW MODE OF REPRESENTING 



*, Ca x) = "-^^^ , which will satisfy the 



'^ ' '' a v) X '' 



conditions of the problem, and equation (2) will 

 become, 



2 Y = f(x) + f,(.r) + ^-^ I f(.^-) - f.W ] 

 I a — X \ I a — X \ 



= [ 1 +7rT^) ^(•^•) + [ ' — «n;r^)^'(-^) (')■ 



so that, whatever value is given to cT, this formula, 

 which is called the equation to the system, will 

 give the corresponding value of Y. 



(9). The function ° ~ ^ , expressing the re- 

 markable property above stated, may possibly 

 lead to some interesting results in the application 

 of analysis to geometry ; in consequence of this 

 we shall call it the operative function. 



The quantity a v> a; is necessarily a positive 

 quantity, without any consideration as to which 

 of the two quantities a or .r exceeds the other. 

 The signs plus and minus, denote operations the 

 reverse of each other, and when they are used in 



