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Full text of "Scientific Papers - Vi"

228       SOME  CALCULATIONS  IN  ILLUSTRATION  OF FOURIER'S THEOREM
The integrals in (2) are afc once expressible by what is called the i integral, defined by
and if the sine-integral were thoroughly known there would be scai anything more to do. For moderate .values of 6 the integral may be cs lated from an ascending series which is always convergent. For la values this series becomes useless; we may then fall back upon a descem series of the semi-convergent class, viz.,
Si(6} = ?-cos0\l~1'2

+
1     1.2.3
smtf (-------2ii~"
0"
DT Glaisher* has given very complete tables extending from 6 = I 0 = 1, and also from 1 to 5 at intervals of O'l. Beyond this point he g the function for integer values of 6 from 5 to 15 inclusive, and afterw only at intervals of 5 for 20, 25, 30, 35, &c. For my purpose these do suffice, and I have calculated from (5) the values for the missing inte up to 6 = 60. The results are recorded in the Table below. In each < except those quoted from Glaisher, the last figure is subject to a. s: error.
For the further calculation, involving merely subtractions, I have sele the special cases 7^ = 1, 2, 10. For fc: = 1, we have
<() = Si (0 + 1)-Si (0-1)......................0
6	Si(fl)	6    Si (6)		0	8i(0)	e	Si(6>)
16	1-63130	28	1-60474	39 '. 1 -56334		50	1-55162
17	1-59013	29	1-59731	40   1-58699		51	1 -55600
18	1-53662	30	1-56676	41   1 '59494		52	1 -57357
19	1-51863	31	1-54177	42 j 1-58083		53	1-58798
20	1-54824	" 32	1-54424	43	1 -55836	54	1-58634
21	1-59490	33	1-57028	44	1-54808	55	1-57072
22	1-61609	34   1-59525		45	1-55871	56	1-55574
23	1-59546	35   1-59692		46	1-57976	57	1-55490
24	1-55474	36	1-57512	47 j 1-59184		58	1-56845
25	1-53148	37	1-54861	48 i 1 -58445		59	1 -58368
26	' 1 -54487	38	1 -54549	49   1-56507		60	1-58675
27	1-58029						
In every case $(#) is an even function, so that it suffices to consid positive.
* Phil. Trans. Vol. CLX. p. 367 (1870). 207.hich m is written for 27dS cos fa. It is to be observed that in this solution there is no limitation upon the value of R if (K  I)2 is neglected absolutely. In practice it will suffice that (K 1) R/\ be small, X (equal to 2?r/&) being the wave-length.