206] SHEET OF DISCONTINUITY. 557 



The integral of this equation is given by 



267) l os <^ = -- + const., = ', 



The last indicated quadrature cannot be effected except by development 

 of the integrand in series, but if we take for the lower limit the 

 value zero, we may express u in terms of the so-called error -function, 

 occurring in the theory of probability, 



268) 



Tables of the values of- Erf(x) have been calculated, and are found 

 in treatises on probability. (Lord Kelvin reprints one such on p. 434 

 of Vol. 3 of his collected papers.) Since the integrand is an even 

 function of x, it is evident that Erf(x) is an odd function of its 

 upper limit x. It may be easily shown that the definite integral 



between zero and infinity has the value -^p so that putting x* = -j> 

 and adding a constant, we have 



269) 



This determination of the constants makes, for all positive values 

 of y and for t = 0, u = u^ (the upper 

 limit being -f oo), and for all negative 

 values u = u 2 , thus giving the dis- 

 continuity required at y = 0. For all 

 other values of t however, no matter 

 how small, the values from the negative 

 side run smoothly into those on the 

 positive, showing how the discontinuity 

 is instantly lost. This is shown in Fig. 171, in which successive 



curves show values of u at times equal to 1, 2, 3, 4, 5, 6 times 

 Differentiating by the limit, we find 



270) fs _4.|. 



