94, REPORTS ON THE STATE OF SCIENCE. 
The Further Tabulation of Bessel Functions.— Report of the Committee, 
consisting of Professor M. J. M. Hitu (Chairman), Dr. L. N. G. 
Fiton (Secretary), and Professor ALFRED LopGE. 
APPENDIX.—Keport of the Committee appointed in 1906 to consider the Further 
Calculation of Bessel’s Functions . : . ; . page 95 
THe Committee were appointed for the purpose of continuing the 
investigation of the formule (see Report for 1906, pp. 494-498) relating 
to the semi-convergent series for the 7‘” Bessel function, viz., 
J,(2)= n/a . Roos (e+a—"—n"), 
where R?=P? + Q? 
_, ,1 4n?—1 1.3 (4n?—1)(4n?—3?) 
Ege Biayar Rama i TT 
the values of P and Q being 
pay —(4—=1)(4n?—3%) | (4n®—1) . . . (4n?9—79)_ 
1.2. (8x) T.2.3.4(8n)" 
qa t= 1 _ (481) Ant —3")(ant— 5") 
8a sone Ra ey = a5 
and a being sin, 
R 
- : ; : /2 : 
with a view to tabulating the logarithms of R and R N / — for different 
T 
values of x and n, and also tabulating values of a. The values of 
2 
R A “~./a give the amplitude of the function, and the values of a 
T 
are needed to give its phase, 
da. 1 
dx R? 
a somewhat unsatisfactory series for a, viz. (writing 8h =4n?—1) 
Hf fess nh oe (5i3—190K2 ee 
ge ~3)_, b(h es 15) _, (5k! 190K? + 807k- 630) 
xv 6x° 102° 56a 
and lastly 
The formule relating to a are Rsin a=Q, and 1+ leading to 
sec (a,41—4,)=R,R,.41; 
which depends on the identity 
P, Pit Q,Q.=1, 
an identity which was obtained by induction, and needed to be verified 
by a deductive proof. This proof has now been obtained by Professor 
Hill, and is appended. 
The Committee have calculated a series of values of log R and log 
R : , which are here printed (see folding Tables I. and IT.). 
T 
They have also calculated values of sin ay with a view to obtaining 
values of a,, ao,... by use of the equation sec (a,,,;—a,)=R,R,,1. 
The values of a,, ao,... they hope to give in a subsequent report. The 
