148 EVALUATION OF CERTAIN DEFINITE INTEGRALS. 



I conceive that this remark is general, and if so we may dif- 

 ferentiate on both sides with fractional indices. Let the index 

 be p, then as 



d* cos zx = x p cos \zx + p ^ J dz p , 

 the first side of (4) will become 



and the second will be 



Thus, we get 



....(6). 



If we take w = m this is equivalent to Laplace's general 

 formula (p), at p. 168 of the Theorie. 



The method of this paper leads to some elegant results when 



/CO 

 e^dx, but it is enough 

 j 

 to point out this application, which involves no difficulty what- 



ever. 



