(ii) from the continued fraction 
Jys(t)_ 21 et 1 
Jy(z) ae) Av+1)_ 2&v+2) * 
x xz 
The ratio of two Bessel functions whose orders differ by unity served as a check 
on the calculations derived from the recurrence formula. When z=14, for example, 
222 REPORTS ON THE STATE OF SCIENCE, ETC. 
: 
the ratio of J20(x) and Js1(z) is : 
16793 8041 45216: 
From (i) J2a(%)= -+ 0-1499 8890 40384: 
and Taa(z)= + 0-0893 1204 79085: 
Dividing J29(x) by the above ratio. 
Jai(z)= + 0-0893 1204 79085: 
The succeeding ratios are : 
1-8694 8979 649: 3-6667 67: 
2:0506 3821 112: 3°8196 50: 
2-2253 7835 892 3°9717 5: 
2-3953 3130 42 41231 7 
25615 8707 57 4-2739 : 
2:7249 1181 7: 4-424 
2-8858 6335 0: 4-57 : 
3-0448 5964 4:72: 
3°2022 2140: 4:8: 
3°3581 997 5-0 
3-5129 945 5-1: 
The calculations were, of course, carried out in the reverse direction. The last 
entry in the above table is the ratio of J as (x) and J 73(a). The relation 
sin vv. 
Ty) yan()+ITy_s(@\T_y(e) = 7 
was employed at intervals to check the results in each table. 
A table of Fresnel’s Integrals S(x) and C(a) to twelve places of decimals was con- 
structed from the values of J»,,, (x) for the first twenty integer values of x by the 
as 
method employed by Lommel for this range of the argument, viz. : 
C(x) = I(x) + J5(x) + J9(t)+JSis(z) +--+ 
S(x) = Ja(x) + Jz(x) + Jaga(z) + Jis(z)+...- 
A number of errors have been discovered in the values of these integrals as 
published in collections of mathematical tables. 
cm 
2y Jv (1) | 2y Jy (1) 
—1 +0-4310 9886 8018 +13 +0: 571 0409 
+1 +0-6713 9670 7142 +15 - | +0: 38 2197 
+ 3 +0-2402 9783 9123 |}. =417 +0- 2 2552 
+ 5 +0: 494 9681 0228 | +19 | +0: 1190 
+ 7 +0. 71 8621 2019 eet -++0- 57 
+9 +0: 8 0667 3904 +23 +0- 2 
+11 +0: 7385 3119 | 
