LAPLACE. 479 



895. There is one proposition given here which is not repro- 

 duced in the Theorie...des Proh., but which is worthy of notice. 



Suppose we require the value of \ydx where y = x^ {\ — oc-y, 

 the integral being taken between assigned limits. 



Put 2^— ^ ^^^ 2' ~ ' ^^^^ ^^^ 



a "a 



1 r7x 



Then, by integrating by parts, 



lydx= \uzdy = c,?/^ — a lydz (1), 



f 7 f dz y dz [ d [ dz\ , 



so that 



/y^x = «^.-a>|+./j;yi (.J)^.. (2). 



Now y vanishes with x. Laplace shews that the value of 

 \ydx when the lower limit is zero and the upper limit is any 



value of X less than , is less than cyz and is fjreater than 



1 + /A ^ * 



dz 

 ayz — o?yz — -; so that we can test the closeness of the approxi- 



mation. This proposition depends on the following considera- 



dz . . . 1 



tions : -^ is positive so long as x is less than , and there- 



fore \ydx is less than c.yz by (1); and — [^ -f) is also jDositive, 



r dz 



so that \ydx is greater than o.yz — (^yz -^ by (2). For we have 



a? (1 — cr) 



z = 



l-(l+yu)a;' 

 and this can be put in the form 



