476 

 in which 



" For the case y^-a, we must change ts*, in (B), to 

 cr^= \ dx\ dt Ptxl 



Jo Jo 



and for the case ;/ z: b, we must change it to 



" For values of ?/ > 6, or < a, the second member of the 

 formula (B) vanishes. 



"If Far, although finite, were to receive any sudden change 

 for some particular value of // between a and 6, so as to pass 

 suddenly from the value f^^ to the value f\ we should then 

 have, for this value of 2^, 



\ c?A dx Vtx-ty Fa; = w' f' + z«r'' f'\ 



By changing p^; to cos x, we obtain from (B) the celebrated 

 theorem of Fourier. Indeed, that great mathematician ap- 

 pears to have possessed a clear conception of the principles 

 of fluctuating functions, although he is not known to have 

 deduced from them consequences so general as the above. 



" Again, another celebrated theorem is comprised in the 

 following : — 



Fy Z:?3— ^ ^M J a ^^ ^"^ "^ ^ ) J ^^ Q^-2/'n ^^) ' (^) 



in which, the function q is defined by the conditions 

 Qxjn \ dxVxZzK dx Pa^; 



J o J2nx~x 



y \s> a, < J; and no real root of the equation 



s 



rfo; Px = 0, 



except the root 0, is included between the negative number 

 a^y and the positive number b—y, nor are those numbers 



