ON THE FURTHER TABULATION OF BESSEL, ETC., FUNCTIONS. 73 



At Quinquisection, 



Region 



(c+3)(c"-4c-l) 

 20cj (c^+l)* 

 (3c-l)(c--4c-l) 

 20c}(c- 1 +lV 



r(c+i)(c-i)ni 



L 32c ! J 



r(c+i)' (e-im 



32c : < J 



I. 



OD>C> V5 + 2 





1 

 ,-= zn?K 



k' 



@ *K=C (18) 

 K=D (36) 



00 

 06 



II. 



V5 + 2>C>1 



zn'K 



zn9K 



0iK 

 1>0 

 0?K 



aV ^F= ^D(18) 



III. 



1>C> 



Vb— 1 



HiK 



=A(18) 



©0 



= -/«'! 



k'C(36) 



HK 

 H|K 

 HK 

 =B(36) 



A(54) 



and so on, as in ' Phil. Trans.,' A, 1904, p. 264. 

 We find that 0=45° for 



c=/^?)^+(^^|=l)*+5i+1^5'184 in Region I, 



c =2 cos 18°+2 cos 36°=3-5202 in Region II, 



c = ( 5 r^)^ + ('^ 2 ± 1 )*_5i + i = o-620 in Region III, 



and so obtain an independent verification of the entry in the Tables 

 calculated from a series. 



These verifications at the Division value are indicated in the Table 

 by round brackets ; values in heavy type have been completely verified. 



Other Division values can be calculated algebraically to serve as a 

 check at any stage of the calculation, and are analogous to the surd 

 values of the circular functions of 45°, 30°, 60°, and the multiples of 18° 

 and 15° ; and it is possible to calculate all the tabular values for every 

 3° from an algebraical formula, by a method analogous to Euclid's con- 

 struction of the quindecagon. 



The entry is given in a Table to every 3° of the quadrant of K, and to 

 four decimals only, as calculated from the first two terms of a q series. 

 Discrepancy is apparent, and more terms are required at a high modular 

 angle, 6 = 75°, as is seen by comparison with the exact numerical value 

 in brackets. 



But the Table is put forward for criticism of the arrangement, and 

 further calculation is reserved. 



The Committee invite criticism, and they are desirous of obtaining a 

 suitable grant from the Association for the expense of their computations 

 on these lines. 



Part II. — Bessel Functions. 



During the year the Committee have been fortunate in securing the 

 valuable help of Mr. J. R. Airey, and they are desirous that his name be 

 added to the Committee. Mr. Airey has calculated, to seven decimal 

 places, the values, for a large range of the argument, of the functions 



