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these examples the procedure will be understood.   When n is even, Differentiate (59) (n— 2) times, thus obtaining
dP \            n  2   r°° d(K
——1M = (—1)2-—       —sin rao sin91 lx.  .........(63)
dr J              Trln Jo   a?
hich smnlx is replaced by the series containing cos nix, cos (n — 2) lx, ... ending with a constant term. When this is multiplied by sin rx, we sines of (r ± nl)x, {r + (n — 2) 1} cc, ... sin roo, and the integration can be ted. Over the various ranges of 2Z the values are constant, but they .ge discontinuously when r is an even multiple of I. The actual forms r.Pn/dr can then be found, as already exemplified, by working backwards r>nl, where all derivatives vanish, and so determining the constants of ^ration as to maintain continuity throughout. These forms are in all 3 algebraic.
tVhen n is odd, we differentiate (??. — 3) times, thus obtaining a form lar to (60) where n = 3. A similar procedure then shows that the result constant values over finite ranges with discontinuities* when r is an multiple of I. On integration the forms for dPn/dr are again algebraic.
'. have carried out the detailed calculation for n = 6. It will suffice to :d the principal results. For the values of
fa. J
:nd for the various ranges :
(r<2Z),  -20;    (2Z<r<4Z),  +10;
(4Z<r<6Z),  -2;    (6Z < r),  0. on integration for
r dr ) '
5r4 ~
- 6Zr» + 30/2r2 - 56ZV +
(4Z - 6Z)f -      + 2Zr» - 18ZV8 + 72Z'r - 108Z4, 1^1
(r > 6Z)        0.
™ ), since the integrals
There are, however, no discontinuities in the value of
u   .  9(r-nl)a      f00 dx  .  9{r-(n-2)l}x            ,. ,            .    .,         ..    ,        .     ,.
sm2 -—„——,   I     —5 sin2-----———i-, etc., which appear m the result when n is odd,
i                   £             j o   x                         &
intintious for all values of r (cf. the solution for -— on p. 621).
The result for (4Z-61) maybe written --^ (6Z-?•)*.   And in general, when (?z-
2 ,-----y— (nl-r)*-2, whether n be even or odd.    W. F. S.]
f(:4 appears to have been devised to meet the case when s = 0 and the integral,, though finite, does not converge to a definite value when x = <x>. If, however, ,s < -1, or <0, respectively, according as the cosine factors in (38) do or do not produce a constant term, the integral (38) has been shown to be finite; it is -also convergent; and the integrals obtained by omitting before each successive differentiation the factor to be differentiated, viz.': cos (%ir-rx-lqir) where g<p, are also finite (cf. Todhifnter's Integral Calculus, 1889, Arts. 214, 284). In these circumstances it would appear that-_(38) is itself valid, and that it is unnecessary to consider the integral obtained by "the next following operation" (s =p + f - %ri). It would seem then that the above considerations are sufficient to justify the differentiation by which 0,t is obtained (jp = 0, s- -f), and a fortiori that for <fe (# = 0, s=-2), etc. W..IY.S.]elative index is then much restricted, the