n-A] WAVE ANALYSIS. 339 



APPENDIX I. 



ORIGIN AND PROOF OF METHOD 



ii. Origin. Many* methods have been used for analyzing a 

 complex wave, either accurately or approximately, but until Rungef 

 devised a simplification the accurate methods have been exceedingly 

 laborious, sometimes involving hundreds of multiplications. Runge 

 found that by combining terms the number of multiplications can be 

 much reduced. 



The method here given is based:}: upon that of Runge, being further 

 simplified by assuming that the negative half-wave is a repetition of 

 the positive, i. e., that there is no constant term and that even har- 

 monics are absent (see Appendix II.). No assumption, however, is 

 made that the initial ordinate is zero (v = o); when analyzing a 

 wave for which the ordinates are obtained by the method of instan- 

 taneous contact such a limitation adds to the labor and reduces the 

 accuracy, for it involves plotting the curve according to the ordinates 

 found by experiment and then measuring from the curve a second set 

 of ordinates, beginning with zero, in order to make the analysis. 

 Evidently it is more accurate, as well as easier, to base the analysis 

 upon the original set of ordinates. 



12. Number of Ordinates Used. If m ordinates in one half-wave 

 are used, the method determines the harmonics to and including the 



* ( ua). For a description of some of these, see Sir William Thomson, 

 Proc. Roy. Soc., XXVIL, 1878, p. 371 5 J- A. Fleming, Lond. Elect., Jan. 

 22-9, 1892 ; J. Perry, Lond. Elect., Feb. 5, 1892, and June 28, 1895 ; S. Ber- 

 son, Eel. Electrique, 1.5, 1898; Houston and Kennelly, Elec. World, XXXL, 

 1898; Michelson and Stratton, Am. Jour. Sc., 5, 1898; A. S. Langsdorf, 

 Physical Rev., XIL, 1901, p. 184; Fischer-Hinnen, Elektrotech. Zeitschrift, 

 May 9, 1901 ; S. M. Kintner, Elec. World, XLIIL, 1904, p. 1023 ; J. Harri- 

 son, Engineering, Feb., 1906, p. 201 ; P. M. Lincoln, Elec. Journal, V., 1908, 

 p. 386; C. S. Slichter, Elec. World, July 15, 1909; C. A. Pierce, Elec. 

 World, Apr. 13 and Oct. 21, 1911. 



t Zeitschrift fiir Mathematic und Physic, 1903, Vol. XLVIII., p. 443. A 

 discussion of Rnnge's method is given by S. P. Thompson, Proc. London 

 Phy. Soc., Vol. XIX., p. 443, and by E. B. Tuttle, Iowa Engineer, Sept., 

 1906. Thompson introduces the condition that yo = o. 



$ Runge's method is followed in combining and tabulating results but not 

 in developing the equations nor in determining phase angles <f> and a. 



