376 Mr Freeman, Value of the least root of an equation. [Apr. 19, 



It was therefore obvious that no more than five terms of the 

 series had been employed in M. Largeteau's calculation. It seemed 

 desirable to push my own approximation further, and after some 

 trouble I found 1'4458 to be a much closer approach to the root. 

 This value of x makes the algebraic sum of the first eight terms 

 of the series to be 0-0000093. 



Proceeding then from the number 1'4458 by Newton's method 

 of approximation applied to the first eight terms of the series I 

 at once found the further approximate value of the root to be 

 1-4457963. 



By way of verification I calculated the value of the first eight 

 terms of the series corresponding to this number, and can write 

 down the results thus : 



+ 



1-0000000 

 •5225817 

 -0075859 

 •0000176 



1-4457963 



•0839496 

 •0004387 

 •0000005 



= + 1-5301852 - 1^5301851 



= 0^0000001, the corresponding sum of the series. 



It will be noticed that the value of the eighth term is numeri- 

 cally only a 5 in the seventh place of decimals ; hence I consider 

 that we can with confidence rely on the value of the least root of 

 this important equation being very accurately expressed by the 

 number above given, namely 1-4457963. 



I may remark that the table in Lommel's BesseVschen Func- 

 tionen is not sufficiently extensive to be of much use in the fore- 

 going investigation. 



But there is a table of the first ten roots of the equation 



2^~^ F74' ~ 2' . 4' . 6' "*" "^ ' 



that is Jq (x) = 0, given on p. 186 of a memoir by Professor 

 Stokes, On the numerical calculation of a class of definite integrals 

 and infinite series (Camb. Phil. Trans., Vol. ix.). 



We have there given the first root (0'7655)7r. 



Now (1 (0^7655 7r)f = 1-4458, 



which is the figure from which my present approximation proceeds. 

 This first root was obtained by Professor Stokes by interpolation 

 from a table given by Sir G. B. Airy. The remaining part of the 

 table was calculated by methods given in the memoir itself. 



