476 



in which 



&=\ _ x dx \ g dt p te - 



" For the case yz=.a,vie must change z<f, in (B), to 

 sr x =\ dx\ dt p to ; 



Jo Jo 



and for the case y — b, we must change it to 

 w" — \ dx \ «fc p te . 



" For values of y > b, or < a, the second member of the 

 formula (B) vanishes. 



"If f x , although finite, were to receive any sudden change 

 for some particular value of y between a and b, so as to pass 

 suddenly from the value F vv to the value f\ we should then 

 have, for this value of y, 



\ dt\ dx Ptx-ty f x = & f' + sr" f . 



J 6 J a 



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 : — 



Vy — lS- 1 P \\ dx F*-4- S \ dx Q x _y, n F*J , (C) 



in which, the function q is defined by the conditions 

 Qxtn \ dx p x zz \ dx p x ; 



J o Jinx — x 



y is > a, < b ; and no real root of the equation 



r*x 



\ dxv x = 0, 



except the root 0, is included between the negative number 

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



