36 



Mr. W. Steadman Aldis. On the Numerical 



For instance if x = zeV.i^) ^ ^^j^ 

 then = Po + Qo«', say, 

 where Po = ^o-^i + ft- Qo = ^2-ft + /3io- 



Thus the values of Po and Qo are easily deduced, and, therefore, 

 that of loixi^). 



The same process gives the value of Jo(««*), for, 



since Jo(«) = Pq- P2 + l^i- 1^6+ 



it is easily seen that 



Jo(xi^) = po^P^i-P^ + PJ^ + Ps- = Po-Qo^ (12). 



The values of Po and Qo are tabulated in the Eeport of the British 

 Association for 1893, to nine places of decimals for intervals of 0*2 of 

 a unit. Table II at the end of this paper gives them for the same 

 number of places, and for the same intervals as have been used in the 

 calculation of the K and G functions. 



Pq and Qo are denoted in the Table II by X and Y in accordance 

 with the notation adopted by the Committee of the British Associa- 

 tion, negative values being denoted by the use of old numeral type. 



7. Assuming the accuracy of the values used for the quantities ^, 

 an accuracy guaranteed by the tests to which the Tables for I and K 

 in the former paper have been subjected, the relation between I and J 

 gives a very easy check for detecting and correcting any mistakes in 

 addition or copying figures in finding the values of J. 



Thus 



571 = 00 



^ Pim + ^P4m+2 



m = 



In finding Jo^^), ^P^m and ^Pim+2 are separately computed by 

 addition of alternate terms frOm lo{x), and the smaller sum written 

 down below the larger. In all cases in Table I the sum of these has 

 first been taken, and the agreement or disagreement of this sum with 

 the known correct value of lo(-i') has shown either that there was no 

 mistake, or has revealed where such mistake was committed, and 

 secured its correction. 



A similar test of accuracy in finding J 1(1) is derived from the known 

 values of I] (a;). 



In like manner, since 



-^Piin ^ "^Psm + ^Psm + ii 



and 



X === Po ^ ^Psm ~ ^Psin + i) 



