134 Mr. W. Williams on the 



19. Hence, finally, if F is finite, single-valued, and con- 

 tinuous between + 7r, or, if not continuous, has only a finite 

 number of finite discontinuities, and where continuous is 

 differentiable, then Fourier's series is convergent, and tends to 

 the limit F(a?) for all values of x except those corresponding to 

 the discontinuities and the values + tt, 4- 3tt, &c. Tne value of 

 the series at a point of discontinuity is -£[F(# — 0) +F(^? + 0)], 

 the mean of the values to which the function tends when ap- 

 proaching the discontinuity from either side ; and its value at 

 + 7r, &c, is J[F(7r) +F( — 7t)], the mean of the values of the 

 function at the two limits. 



Ill, 



20. The simplification in the above method of evaluating 

 the integral S w consists in having first proved that the two 

 portions A and C taken respectively between the limits — it 

 to —h, and h to it vanish when n=co however near to the 

 value zero ive take the ordinate s +h, so that the value of the 

 integral depends only upon the value of the infinitely thin 

 strip B taken between ±h. S^ is therefore independent of 

 the values of F(0 + #) outside the strip B, and consequently 

 is the same as if F(0 + «ip) remained constant throughout and 

 equal to its mean value F(#) within B. That is, S^ =F(^). 



21. The vanishing of A and C when w=.oo depends upon the 

 fact that the function integrated, namely %(i#) sin (2n + l)±d 

 has an infinite number of finite oscillations (that is, oscillations 

 of finite amplitude) between — tt and —h, and between h 

 and tt. Hence, since the number is infinite and the ampli- 

 tudes finite, neighbouring oscillations differ infinitely little 

 from each other, and therefore the area included between the 

 ordinates — it and —A, or h and 7r, and the portions of the 

 function and the axis of 6 intercepted by them is infinitely 

 small. In other words, the mean value of the function from 

 — 7rto —A, and from h to 7r is zero, and therefore the integral 

 of the function between the same limits is also zero. But the 

 function itself is not zero : it is merely indeterminate, — the 

 oscillations being, as it were, too fine-grained to be traced 

 individually. The transformation (27i + l)J0 = <£, however, 

 resolves these oscillations, however fine-grained they may be, 

 into oscillations of finite period cutting the axis of at equal 

 intervals 7r. We are therefore enabled to deal with each 

 individual oscillation instead of with the oscillations as -a 

 whole, and so to determine the precise effect of each upon the 

 value of S w . 



22. If we break up the portions A and C of the integral S„ 



