564? LAPLACE. 



then X=F6^^. 



Substitute in (1) and we obtain 



P= — j F cos (?/ — ex) sin vx — 

 + / F sin (?/ — ex) sin 9;^ — . 



TT J -00 ^ 



The elements in the second integral occur in pairs of equal 

 numerical value and of opposite signs, while the elements in the 

 first integral occur in pairs of equal numerical value and of the 

 same sign. Thus 



2 f" dx 

 F = —j Y cos (2/ — ex) sin 7]x — (2). 



Since each error is supposed to lie between a and h we have 



I fi (^) ^^ = 1- 



-^ b 



Hence it follows that pi = l when x = 0; and we shall now 

 shew that when x has any other value pi is less than unity. 



For pl = \j fi{z) cos jiXzdzY +\ fi(z)smyiXzdz[ 

 fa ra 



that is Pi^ = fi (^) COS 7^ xz dz fi (z) cos 7^ xz dz' 



-lb J b 



ra ra 



+ 1 fi{z)^iriyixzdz j f {z') smji xz' dz 



ra ra 



""] J fi{^)fi(^^')^osry,x(z'-z')dzdz; 

 and this is less than 



that is less than 



J b J b 



that is less than unity. 



12 



