50 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
Table I.—Values of Acc 2 for Single Refracting Surface. 
M. 
cTT. 
First order. 
Second order. 
Fractional 
formula. 
True. 
10 
0‘5 
-33-35293 
- 107-6252 
+ 27-18563 
+ 26-66794 
10 
0-25 
- 8-33823 
- 12-98025 
- 18-81008 
- 18-85273 
2 
0-25 
- 0-250000 
- 0-378906 
- 0-516129 
- 0-527526 
2 
0-125 
- 0-062500 
- 0-070557 
- 0-071749 
- 0-071781 
0-5 
0-5 
- 0-015625 
- 0-016541 
- 0-016598 
- 0-016611 
0-5 
0-25 
- 0-003906 
- 0-003963 
- 0-003964 
- 0-0039133 
0 
0-5 
- 0-166667 
- 0-174769 
- 0-175183 
- 0-175809 
0 
0-25 
- 0-041667 
- 0-042173 
- 0-042179 
- 0-042189 
- 1 
0-5 
- 1-000000 
- 0-947500 
- 0-950119 
- 0-956680 
- 1 
0-25 
- 0-250000 
- 0-246719 
- 0-246761 
- 0-246838 
Table II.—Values of Sin a 2 for Single Refracting Surface. 
M. 
CT. 
First order. 
Second order. 
Fractional 
formula. 
True. 
10 
0-5 
+ 0-0250865 
+ 0-0454644 
+ 0-0576348 
+ 0-0610770 
10 
0-25 
-0-0078936 
-0-0072568 
-0-0071911 
-0-0071828 
2 
0-25 
-0-2187500 
-0-2065430 
-0-1987180 
-0-1978219 
2 
0-125 
-0-1210938 
-0-1207123 
-0-1206710 
-0-1206691 
0-5 
0-5 
0-2539063 
0-2541962 
0-2542195 
0-2542230 
0-5 
0-25 
0-1254883 
0-1254974 
0-1254975 
0-1254974 
0 
0-5 
0-1805556 
0-1817130 
0-1826667 
0-1827294 
0 
0-25 
0-0850694 
0-0851267 
0-0851286 
0-0851291 
- 1 
0-5 
0-1250000 
0-1299375 
0-1311526 
0-1314354 
- 1 
0-25 
0-0531250 
0-0532793 
0-0532873 
0-0532884 
It appears from the above that the fractional formulse are not merely equal, but 
appreciably superior to the second order formulse, and this not merely in cases such 
as those of the three first entries in Table I., in which the convergency of the series 
for Ax 2 is either absent or slow, but in every case where the fractional or second order 
formulse differ sensibly from the true value. (Clearly a divergence of 1 in the last 
place cannot be claimed as significant, for the last figure in Tables I. and II. is probably 
not correct within +2, in some cases.) An estimate of the range of the formula can 
be obtained from the fact that in the cases, vy = 0‘5, = 10 and 2, the angles of 
incidence were 52° 34' and 48° 35' respectively, and, for the other values, angles of 
incidence of 20° and 30° are quite common. 
In view of this the accuracy of the results is surprising and, from the point of view 
of the further applications of the method, most encouraging. 
