538 Mr. stokes, on THE CRITICAL VALUES OF 



of x' two quantities lying as close as we please to x, and therefore so close as to exclude all values 

 of w' for which /(■»') alters discontinuously. Let ^ = l - /;, x = x + ^, expand cos — by the 

 ordinary formula, and put /(a?') =f{x) + R- Then the limit of (7) will be the same as that of 



the limits of ^ being as small as we please, the first negative and the second positive. Let now 



(^-■■■) = r'. 



so that — ^ is ultimately equal to — , that is to say when g is first made equal to 1, and then the 

 di, 



limits of ^, and therefore those of ^', are made to coalesce. Let now G, L be respectively the 

 values of (l - 1 A 



\ dp 



greatest and least values of (l - 1 A) - — |, {f(x) + R\ within the limits of integration. Then if 



and 



we observe that I — — ^ = tan"" y + C, where tan"' denotes an angle lying between - 



*/ /i "T* C '^ 



putting - ^,, ^2 for the limits of ^', we shall see that the value of the integral (8) lies 

 between 



G ftan-' I + tan-' ^] and L [tan-' ^ + tan"' |'j : 



but in the the limit, that is to say, when we first suppose h to vanish and then ^, and ^j, G^ and 



L become equal to each other and to -f{x), and tan"' ^ + tan"' ^ becomes equal to tt. Hence, 



f(x) is the limit of (7). 



Next, suppose that the value of x which we are considering lies between and a, and that 

 as x passes through it/(*') alters suddenly from M to N. Then the reasoning will be exactly as 

 before, except that we must integrate separately for positive and negative values of ^', replacing 

 f(x) + R hy M + R in the latter case, and by iV + -ff' in the former. Hence, the limit of (7) will 

 be ^(M+ N). 



Lastly, if we are considering the extreme values x = and x = a, it follows at once from the 

 form of (7) that its limiting value is zero. 



Hence the limit to which the sum of the convergent series (5) tends as g tends to 1 as its limit 

 is/(.r) for values of a; lying between and a, for which /(.r) alters continuously, it is i(M + N) 

 for values of x for which /(.r) alters suddenly from M to N, and it is zero for the extreme values 

 and o. 



5. Of course the limiting value of the series (5) is /(O) and not zero, if we suppose that g 

 first becomes 1 and then x passes from a positive value to zero. In the same way, if f(x) alters 

 abruptly from M to JV as x increases through .r,, the limiting value of (5) will be M if we suppose 

 that g first becomes 1 and then ,v increases to a;,, and it will be N if we suppose that g first 

 becomes 1 and then x decreases to .t, . It would be futile to argue that the limiting value of (5) 

 for a? = is zero rather than /(O), or /(O) rather than zero, since that entirely depends on the 

 sense in which we employ the expression limiting value. Whichever sense we please to adopt, no 

 error can possibly result, provided we are only consistent, and do not in the course of the same 

 investigation change the meaning of our words. 



