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Full text of "Scientific Papers - Vi"

394-                           ON LEGENDRE'S FUNCTION Pn (Q),                          [404
If we take out a further factor, eime, writing
u = vsm~^0 = weimesm~^6,   .....................(3)
of which ultimately only the real part is to be retained, we find
dhu       .   dw        w                                                   ,.v
d& + 2mldd + fStf-e = °.........................(4)
We next change the independent variable to z> equal to cot 6, thus
obtaining
dw     -i (,_         d*w         diu    w
-7-= 5— u cte     2m (v
From this equation we can approximate to the desired solution, treating m as a large quantity and supposing that w = \ when 0 — 0, or $ = ^TT.
The second approximation gives
dw         i           ,                   .,      *'# •
_ _  —       whence     w = l— -5— . rf5        8m                                   8m
After two more steps we find
~ i      •  f JL _     9_J _   9-g2    ,    75^a                      p,
W~       ^V8m    128mV    128m2+1024ms.............l ;
Thus in realized form a solution of (1) is
^^)oo8(«K9 + 7)
foot 5    9cot'0    75 cot3 0)   .   ,   Q       ']           /17N
+ ^-5--------T-JJ5—- — T^oT—rh sin (me' -f 7)   ; ......(7)
(8m      128m3     1024m3]        •          ' _]
and this may be identified with Pn provided that the constants C, 7, can be so chosen that u and du/dd have the correct values when 0 = ^7r. For this value of ^ we must have
-Pn(i7r) = (7cos(£m7r + 7),........................(8)
(dPn/dd)^ = (7 (- m - §L + __!_) Sin (im?r + 7)..........(9)
We may express (dPnld6\   by means of Pn+1 (^-TT).    In general
rfP
Sin2 ^ r7 ^
w LUo (7
so that when 0 = £ TT,
dPn/de = ~dPn/dcQBd = (n + l)Pn+l................(10)
When wis even,(dPn/dd)^v vanishes, and, G being still undetermined, we may take to satisfy (9), 7 = - |TT ; and then from (8)
(7 cos (Jn*-) = Pw (i,r) = (- I)*'1        •"^"^ ,n as equal to unity.    W. F. S.] unexpectedly, that there is a linear acceleration amounting to