STORY. A NEW GENERAL THEORY OF ERRORS. lio 



<f> (x) (which is positive for any possible error x) is at each limit ; * 

 if 4> (x) is not for any Jinite value of x, all values of the error are to be 

 regarded as possible. The results will show that 4> (^) = for x = ± cc, 

 and, in general, (j> (x) has a very small value for any large value of x. 

 We have, then, 



(19) (t>(x) = and <^ (x) = 0, 



so that tlio last two terms of Xq + Sx^, vanish, and, on reversing the order- 

 of integration, by (17) and (9), 



x' x-\-x' 



Xq + 8xo = j cf)' (x') • dx' • I X • (f) (x — x') • dx 



x' x-f-x' 



X' 



= j (Xo + x') </>' i^-') dx' == Xo + xo' ; 



so that 



(20) 8xo = xo'. 



By (10), as applied to both component sources and their combination, 

 we have 



^< {x<) = 4>> {^> + X,') = ^' i^), 



^(x-x') =c{,{i — $' + Xo .- Xo') = xl,(i- ^' — Xo'), 



i> (x) + 8ct> (x) = (<^ + 8<^) a + Xo) = (-A + S^) a - Xo') = 



</' (f - a^oO + H (t - Xo'), 



* The relative frequency ^'(1) of a possible residual | must be positive; if | 

 varies continuously from zero, either through positive or through negative values, 

 the limit | or | must be reached before \p (|) becomes negative ; but if x^/ {(} passes 

 through zero without changing sign as | passes through a certain value, — a value 

 that corresponds to a point of contact of the curve represented by >/ = 4* (f) with 

 the axis of abscissae, — tliat value of | is not necessarily a limit to tlie possible 

 values ; such a value of | could never occur as a residual, althougli values on botli 

 sides of it might occur. However, if the conditions under which the observations 

 are made change continuously in such a manner that the curve y — ^ (|) assumes a 

 form in which it touches the axis of abscissae, this must be the limiting (extreme) 

 form, provided a continuation of the change would produce a curve having negative 

 ordinates in tlie neighborhood of the point of contact; namely, on account of the 

 permanence of the relation (14), the limits J and | cannot change discontinuously. 



