1914] SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM 231 An inspection of the curves plotted from the above tables shows the approximation towards discontinuity as ^ increases. That the curve remains undulatory is a consequence of the sudden stoppage of the integration at & = &a. If we are content with a partial suppression only of the shorter wave-lengths, a much simpler solution is open to us. We have only to introduce into (1) the factor e~ak., where a is positive, and to continue the integration up to as = oo . In place of (2), we have tan-1 <f> (a,) = rdl™~~ {Sin % 0 + 1) _ sin k(x- 1)1 = tan"1 (-—} -Jo « \ a J ...... (9) The discontinuous expression corresponds, of course, to a = 0. If a is merely small, the discontinuity is eased off. The following are values of $(#), calculated from (9) for a = 1, 0'5, 005 : X $(x) X *(*) X </>(") o-o 1 -571 2-0 0-464 4-0 0-124 0-5 1-446 2-5 0-309 o-o O'OSO 1-0 1-107 3-0 0-219 c-o 0-OB5 1-5 0-727 a = 0-5. a = 0-05. X <!> («) X •HT) X </> (») o-oo 2-214 1-00 1-326 2-00 0-298 0-25 2-J73 1-25 0-888 2-50 0-180 0-50 2-111 1-50 0-588 3-00 0-120 0-75 1-756 1-75 0-408 3-50 0-087 X *(*) i X *(*) x 0(*) o-oo 3-041 0-90 2-652 1-20 0-222 0-20 3-037 0-95 2-331 1-40 0-103 0-40 3-023 1-00 1 -546 1-60 0-064 0-60 2-986 1-05 0-761 1-80 0-045 0-80 2-869 1-10 0-440 2-00 0-033 As is evident from the form of (9), <£ (as) falls continuously as as increases whatever may be the value of a. . ' .59525 45 1-55871 56 1-55574