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1914]                          SPHERES  OF SMALL RELATIVE  INDEX                            223
The integral may be expressed by means of functions regarded as known. Thus on integration by parts
rm
{l+m" + (m2 - 1) cos 2m  2m sin 2m}
J n
1 - cos 2m    sin 2m _   1       1 ~ "      ~"2m~ ~~ 2m"2 + 2 '
rm                                                                   (m
(1 + m2 + (m2  1) cos 2m  2m sin 2m\ r Jo                                                                  W
1        fml  cos 2m 7       cos 2m    sin 2m
m                   2m2          m
{1 + m2 + (m2  1) cos 2m  2m sin 2m| -Jo                                                              m
cos 2m ,       m2    m sin 2m    5 cos 2m    5
-------dm + -$- +----~----+ -:------- j.
.' o         m       22       44
Accordingly, if m now stand for 2kR, we get
7 (1 - cos 2m)
Q-iv-i-.yV-./T/.^L/   / a   i    ft 'z   i    t, a \ __                            /      i           ^                          '
sin 2m .  .     . / 4 _ A [m lji_cos_2m ^ ,            ,13
m                     V??!"      /Jo         '^
If m is-small, the { } in (13) reduces to
0 + 0 x m2 + ^ m4, so that ultimately
/^-l)2, ........................(l^)
in agreement with the result which may be obtained more simply from (5). If we include another term, we get
As regards the definite integral, still written as such, in (13), we have
1 cos2m ,        ramj x      xz     xs        [,;,_       i     /o   ^   PV9   \   nfi"\ o        m               Jo    (1.2    4!    6!   '"J
where 7 is  Euler's  constant  (Q'5772156) and  Ci is   the   cosine-integral, defined by
"" cos^t
~.                       ,                                      /1l7v
Ci(flj)=      - du ............................ (17)
^00        U
As in (16), when x is moderate, we may use.. (12)