74 REPORTS ON THE STATE OP SCIENCE. 



previously dealt with by Aldis and by B. A. Smith. The need for more 

 extended tables of these functions has always been felt. 



The tables mentioned in the Eeport of the previous year are not yet 

 sufficiently advanced for publication, and in the meantime it seems 

 desirable to publish Mr. Airey's tables in the present Report, since they 

 are now complete enough in themselves. The whole of the work, of which 

 an account is now given, is due to him. 



Tables of the Neumann Functions G„(x) and Y„(x). 



(A.) 

 The values of the functions G n (x) and G,(x) have been calculated by 

 Aldis 1 to 21 decimal places from x = 0*1 to (K) by intervals of 0*1. In a 

 paper by Michell on ' The Wave-resistance of a Ship,' tables of these 

 functions in the form 



kJ (x)— Y„(x) and kJ 1 (x)—Yi(x), 



where /.• = log 2— y = '11593 . . . 



were given by B. A. Smith 2 from x = , 00 to l'OO and from 1*0 to 103. 

 The calculations were carried out to four places of decimals with an 

 error of one in the last place and possibly of two wnen the value of x is 

 greater than 3 or 4. 



The following values of the G (x) and G^x) functions were calculated 

 from the semiconvergent series 



G„<s)=- Vta { P » 8in ( x -l) + Q » ""("-j) } . 



and the results verified from the relation 



(jiJo — WqJ i ==-. 

 x 



The G functions can als-> be calculated from the J functions, and this 

 provides another method of verification : — 



For large values of x, 



and 



Jo<*)= \/* x ■ [ PoW cos( x- £) - Q„(.r) Bto(a-]) } 



Put Q tl (x)=R u sin O and P n (x)=R cos ti . 



Then 



tan —Q {l (x) / Po (z) and O can be found. 

 Hence 



Go(s)= —J • Jo(«) • tan I x— |+°oJ • 



1 Aldis, Proc. Roy. Soc, lxvi., 1899-1900. 



2 Michell, Phil Mag. [5], xlv., 1898. 



