293 



the upper or lower sign being taken, according as q is posi- 

 tive or negative. Assuming as the definition of r(w), the 

 equation 



V{n) — \ dx cos (x) a;""', 



whether n is positive or negative, and regarding the integral 

 in the second member as a limiting integral of the first or 

 second class, according as n is positive or negative, the author 

 shews that, universally, 



r(n)r(l-w) = -r^, 

 smnn- 



a theorem which is known to be true of F in its ordinary de- 

 finition when n lies between and 1, but not otherwise. This 

 theory is further applied to explain the discontinuity of form 

 which is apparent in integrals, the subjects of which become 

 infinite within the limits of integration, with some other con- 

 nected points. 



The paper concludes with an application of Fourier's 

 theorem to the solution of equations. It is proved that the 

 value V of the definite integral 



v=— C^ C^ dadvi''''-'')^''^-^f{a,vy/-\) 



27r«^-x J-x 



is symbolically expressed by the equation 



d 



dV» 



f a x"" 



provided that =: 0, from which the following theorem is 

 deduced : 



\i f{u) = X and ^{x) be any function of x which makes 



/[^ (*)] ^^^i t^^" 



rf _ 



M'hich may be expanded in the form 



