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Full text of "Scientific Papers - Vi"

360                          ON  THE  THKOHV   OF THK  CAI'IM-AUV   TI'HK                       j.'l!l!
the conatant of integration being dotrn.iiiK'd l.y lh- ronv^iondru.-.' of .r     / = -71-.    Thus
= log (V2 - 1)4V2 -1- ''   - 2 4- N;   4     logU 2 -
''              *      *
giving when -fr is small
where                = log <V2 - D + V2 + 1"K ^ - - -' - {)'usn!<- ......... <lVh
The other e({uation, doriwd from the flat purl of the uri','M-.\ .      dz     // , (ac\     ltn''i.r" /,      '
X^1  S3           5=          /i(        IS?                                I     I     ~"             I  ,        ,,,,.
Y       f/;t;      rt       \/      \ ( 2ir<  )          \'
iu which a;/(t is regarded an large ; >r
UT-                             tt                "ifT.i"       -i"
log     -H 3<K         4-      .      ........

In equations (58), (57) ./ and ^ nrr- t> hi- itii>ntiH<*(l.   < >n
7*    .     tt     a  <^/3r      r~.r      .,     27r.r     't"
-log .-=-:, ..... -'    + rt         4-UK         
a       fe hu      1 + a/2/(     2/- + n    *    ^   n       K
in which wo may put
27r.r    .     2-Trr           f.      r  ir             *2wr    r - .*
~
rm                             i                                                *~*.r        t       ,
Thus         --log,   =   .    *'     4-   '           fu   4-
n          /'o      1 4- a2r          Hr3         Hr     -
in which, since as is nearly equal to r, (r-^|/Hr' tuny UMiuliy U- t)}*ii*ri.ml Also, in view of the mnalhu'HS of  and /y, it in eiim*ly ti'f'?yi;irv i* r'tan the denominator 1 4-a/2r, BO that w may write
;/ = - 0-0809 4- 0-270H " 4- 1 log "y
- 0-8381 + 0'2798,/r 4- 1 log I r/ ). ........... , till I
The effect of the second approximation m the intruthictiou uf tlir -pi term on the right of (GO).+ 4a/3r, and