148 Mr. R. Appleyard on Dielectrics. 



infinite number of maxima and minima values, Eiemann has 

 shown in the paper already referred to that this condition is 

 not sufficient. For, although the integral of the function 

 taken at the point where it is infinite may vanish, this integral 

 when the function is multiplied by cos nx or sin nx where 

 n=co may become infinite. In the first case the oscillations 

 of the function mutually compensate each other, but in the 

 second case the factors sin nx or cos nx may destroy this 

 compensation when w = oo, for the oscillations of the two 

 factors may conspire to produce a resultant function which 

 is infinite without oscillations. The value of the integral in 

 such a case is, of course, indeterminate, and so it is not suffi- 

 cient merely to know that the integral of the function vanishes 

 at the point where it is infinite. 



41. The complete investigation of the convergency of 

 Fourier's series ultimately resolves itself into a discussion of the 

 conditions of integrability and the nature of functions. We 

 thus see that the inquiry leads us to the very foundations of 

 the Infinitesimal Calculus, and in this respect Fourier's series 

 differs essentially from Taylor's. For in the case of the latter 

 series the field of investigation is, at the very outset, restricted 

 for us by the nature of the coefficients, since the process of 

 differentiation limits us to functions of a comparatively simple 

 kind. 



,. 



s 



XII. Dielectrics. By Rollo Appleyard *. 



OME experiments upon the change of resistance of certain 

 dielectrics with the duration of the testing-current, and 

 with the testing-voltage, were described in a paper f which I 

 read before the Physical Society two years ago. In continua- 

 tion of this research some further tests have been made, the 

 principal object being to determine the effect of temperature 

 upon the dielectric resistance. For this purpose, mica and 

 paraffined paper, in the form of condensers, have been 

 chosen. 



The resistances are measured by the " direct deflexion " 

 method, and are expressed in megohms pro microfarad. The 

 testing-voltage is the same throughout all the tests (450 

 volts), and each measurement is computed from the galvano- 

 meter-reading noted after the testing- current has been 

 applied for one minute. 



Two paraffin-paper condensers, each of one microfarad, 



* Communicated by the Physical Society : read May 22, 1896. 

 t "Dielectrics," Proc. Physical Soc. xiii. p. 155, 1895; Phil. Mag. 

 Oct. 1894, p. 396. 



