Scientific Papers - Vi

90
ON THE PROPAGATION OF WAVES, ETC.

The complete solution of (97) in series of ascending powers of as is to obtained in the usual way, and the arbitrary constants are readily determii by comparison with (98). Loramel* showed that these series are express] by means of the Bessel's functions /j, J-i. The connection between complete solutions of (97), as expressed by ascending or by descending se convergent series, is investigated in a second memoir by Stokes f. A rej duction of the most important part of Airy's table will be found in Masca Optics (Vol. I. p. 397).
As total reflection requires, the waves in our problem are stationary regards x. The realized solution of (95) may be written
........................ (106
W being the function of x already discussed. On the negative side, \vhe: numerically exceeds a moderate value, the disturbance becomes insensible.
* Studien iiber die Bessel'sclien Functional, Leipzig, 18G8. t Camb. Phil. Trans. 1857, Vol. x. p. 106.be taken when these quantities are negative, he found when m is positive