LAPLACE. 563 



Let us now suppose that w is indefinitely small, and that X 

 and jx are infinite ; and let 



Xft) = (c + 7;) -ST, 11(0 = (c — rj) tar, 'urO = 



cox. 



The limits of the integration with respect to x will be + 00 . 

 Also we have 



,^ ft) , • 1/1 «^ 

 av = — ax, sm ^a = tt- . 



Thus, neglecting ± - compared with X and /^, we obtain 



P=- Xe-''^^-^ sin 77a'— (1). 



This expression gives the probability that 'stE will lie between 

 (c + 77) t3- and (c — rj) 'ur, that is, the probability that E will lie 

 between c + rj and c — 17. 



Since we suppose co indefinitely small we consider that the 



error at each observation may have any one of an infinite number 



of values ; the chance of each value will therefore be indefinitely 



small. Let 



aft) = a, /3ft) = h, noa — z\ 



and for X in (1) we must put the new form which we thus obtain 

 for the product 



Assume fi (z) cos 7"^ xz dz = pi cos ?\, 



I fi (z) sin ji xz dz = pi sin r^ ; 



then Qi- pi^ 



Let Y= p^p^ p^ ... p8> 



y = rj + r^ + 7-3 + . . . + r, ; 



86—2 



