232 Sir William Thomson on an Alleged Error 



or 



_2£ + l 



-('-■)« 



3 , 



when & is infinitely great. Now this is precisely the degree of 

 ultimate convergence of the coefficients of x 2k , w ik+2 , &c. in the 

 expansion of (1 —x 2 )*. Hence, when x is infinitely nearly equal 



to unity, a is finite, and so also is \/(l —a? 2 ) -j-, or —z. Now 



clearly at the equator (or when a?=l) we must have -jt. = 0, 



because of the symmetry of the disturbance in the northern and 

 southern hemispheres in the case proposed for solution by 



Laplace and Airy. Hence in this case 2 +2 must converge 



&2k 



to zero, and therefore K 4 must have the value given to it by 

 Laplace. 



8. Look now to the degree of convergence obtained by 



Laplace's evaluation of K 4 and verify that it secures —^ =0 



do 



when 0z=±7r. Put 



= tt* (/) 



-2/c + 2 



By this we have 



and (G) resolved for B* gives 



e 



'^'sSr^r (8) 



U + 6 K * +1 



Hence, unless the ratios converge to unity, (8) gives 

 2e 

 R * = k(2k + 3) when * is great (9) 



Now Laplace's determination of K 4 by his continued fraction 

 implies the determination of the ratios by taking R A+1 = for 

 some very great value of k and calculating 



Rj:, R*_u Rj;_2 



by successive applications of (8) with k—1, /c— 2, . . . substituted 

 for k. Hence it gives to the series (3) a degree of convergency 

 (approximately the same as that of the expansion of e x ^ e + e~ x ^ c m 



