108 LOED EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE, 



These examples indicate that the first root is always greater than the value given 

 by the approximate formula, and the successive intervals are slightly less. 

 The actual roots of P 7 are, approximately, 



18 24', 42 9', 66 fr 90, &c., 

 and those given by the Laplace formula are 



18, 42, 66, 90, 



so that the true roots are all a little ahead of those given by the Laplace formula, 

 so long as 6 is less than 90. Beyond 90 the roots are, of course, similarly spaced in 

 everse order. 



NOTE BY LORD RAYLEIGH. 



Professor LODGE'S comparison of P 20 with LAPLACE'S approximate value suggests 

 the question whether it is possible to effect an improvement in the approximate 

 expression without entailing too great a complication. The following, on the lines of 

 the investigation in TODHUNTER'S ' Functions of LAPLACE, &c.,'* 89, shows, I think, 

 that this can be done. 



MACMILLAN and Co., London, 1875. 



