38 



Mr. W. Steadman Aldis. On the Numerical 



9. The well-known sequence laws * 



J„^,(,^) = ??J4^)-J„-i(.:.) (U), 



dJoldx = - Ji (15) 



can be utilised, the former to obtain the values of J2{^i^), ^si^i^) 

 and the latter to give a verification to some extent of the values of 

 Ji{xi^), by means of the formulae given in the writer's paper on I and 

 K, which express dy/dx in terms of a series of equidistant values 

 of y. 



Thus, since 



dJJdx = -Ji, 



replacing x by xi^-, and using the values already assumed for Jo{xi^) and 

 Ji(a;r), it follows that 



diX-Yi)_ .Xi + Yi^_ l+i Xi + Yii^ 



(16). 



:i+Yi) J 



dx J2 ' J2 J2 



Whence dX/dx = -i(Xi-Yi), 



dY/dx = i(X 



By means of the formulae (18), (19), and (21) of Articles 17 — 19 in 

 the paper above referred to, this formula gives a check to the series of 

 values in Table II to a considerable number of decimal places, to 

 thirteen places with the last approximation. 



10. For determining the values of J2{xi^), J3(«^^*)j ... % the sequence 

 law, it is convenient to denote these quantities by the symbol X^ + Y,^^ 



when n is even, and by {Xn + Yni) when n is odd. This will be 



found to avoid irrational multipliers in the successive derivations. 

 Equation (14), putting xi^ for x, gives 



2u 



XI 



The cases of n odd and n even must be separately considered. 

 First let n be odd. The equation gives, remembering that = 

 1 - 2 



Xn + i+Yn + ii = -{l-^)(Xn+Yn^)-(^n-l + Yn-l^). 



Whence, if n be odd, 



Xn+i = - {Xn i- Y;i) - X,i_] 



n 



Yn + l = - {Yn -■ Xn) - Y,)_i_ 



X 



(17). 



