296 Proceedings of the Royal Society of Edinburgh. [Sess. 
K 
1-5 
2-5 
3 
4 
oc 
log aj 
I 
2-61979 
1-09691 
1-22185 
1-39794 
2*92082 
log «2 
1 3-08468 
3-41567 
3-44370 
3-31876 
4-84164 n 
log « 3 
5-81361 
5-94369 
5-82045 
6-91736 
5-13921 
log a 4 
6-59587 
6-45524 
6-10924 
7-23612 
7-61229 n 
log « 5 
7-39883 
8-96928 
8-46991 
9 5540 n 
8-19003 
log «6 
8-21416 
9-49350 
10-72486 
11-8517 
10-83708 n 
log a v 
9*03834 
10-01706 
11-19035 
n 
11-5332 
log «8 
11-86917 
12-54628 
14-84984 
n 
log a 9 
12-70515 
13-07368 
14-25996 
log a l0 
13-54520 
15-60715 
16-9440 n 
log «n 
14-38854 
16-13498 
17-9204 
log a 12 
15*23460 
18-67438 
n 
log a 13 
19-1967 
log «14 j 
21-7511 
log «15 
22-2473 
log “16 i 
24-8548 
log a l7 j 
25-4061 
log «18 j 
27-6267 
§ 14. In each of the functions (13) and (17) the coefficients become 
positive and negative alternately ; and for small values of x, close enough 
to q, the series ultimately diverge. If x is not too small, several terms at 
the beginning of the series converge fairly rapidly, and give a close 
approximation to the result. The following values of the Homer Lane 
Function calculated by means of the numbers given above are correct to 
the last figure shown: — 
©^(•4) = *3159 ; ® , 1 . 5 (-4) =2-133 ; © rs (-35) = *2007 ; ®' 1 . 5 (*35) = 2*464 ; 
© 2 . 5 (-3) =-24186 ; ®' 2 . 5 (-3) =2-0078; @ 2 . 6 (-25) = *13768 ; ©' 2 . 6 ('25) = 2-1448 ; 
©o (-25) = -2093 ; ©' 3 (*25) = 1*922 ; © 3 (-2) =-111 ; ®' 3 (-2) =2‘00; 
© 4 (-25) = -3180 ; ®' 4 (-25) = 1-582 ; © 4 (-2) =*2359 ; @' 4 (-2 ) = 1/695. 
By comparing these results with the corresponding values of the Homer 
Lane Function given in Tables I. . . . IV., we see that the numbers obtained 
by the step-by-step process of § 8 are sufficiently accurate for our purpose. It 
is possible to calculate q and ®' K (q) from the values of ® K (x) and ®' K (x) at 
a point at which they are given with sufficient accuracy by (13) above, by 
means of the approximate formulas used by Homer Lane and referred to 
in § 3 above. It will be seen later, however, that Homer Lane’s calculated 
results agree with those given in the tables. 
§ 15. We can now estimate from the numbers given in § 14 the degree 
of probable error in the values of q and ®'(q) found by the step-by-step 
process. Similar reasoning to that of § 9, by which we show that errors 
arising in the working out of this process do not tend to increase as the 
work proceeds, leads to the following results : — 
The value of q{‘ 2737) in Table I. is correct to the nearest figure, while 
