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Published online by Cambridge University Press: 20 January 2009
The numerical solution of Integral equations with variable upper limits has been investigated by Professor Whittaker. In this investigation the nucleus, supposed to be given numerically by a table of single entry, is replaced by an approximate expression consisting of a finite number of terms, each term involving an exponential or simply a power of the variable, and then the solution is found as an analytical expression from which its numerical values may be computed. The numerical solution of integral equations with fixed limits has been discussed by H. Bateman.
* Proc. Royal Soc., XCIV (A), 1918, pp. 367–383.
† Proc. Royal Soc., C(A), 1921, pp. 441–449.
‡ See, for example, Bond, Proc. American Academy, IV., 1849, pp. 189–203, or Encke, Aslronomische Jahrbuch für 1858.
§ Bashforth and Adams: An attempt to test the Theories of Capillary Action, Cambridge, 1883.
‖ Mathemituche Annalen, XLVI., 1895, pp. 167–178.
** Zeiischriftfiir Math. u. Phys., XLV., 1900, pp. 23–38.
†† Ztitschrift fur Math. u. Phys., XLVI., 1901, pp. 435–453.
‡‡ Phil. Mag., XXXVI. (6th Ser.), 1919, pp. 596–600.
* It must be borne in mind that the generalised Simpson's rule is less exact than formula (2) when we have calculated
* See Vergerio, Annali di Matematica, XXXI., 1922, pp. 81–119.
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