42
Hence, as before, taking three values of y for x, xk, xk 2, the solution is—
(us)......p - * . bg /j°gy«-*>gy»\
logfc	I log 2/a — log ?A I
(116	)......n = lQg y* *
'	xv (kP — 1) log e
(117	)......log B = log 2/1 —nxP log e.
These curves are shewn for Fig. 23 hereinafter, for various values of
n and p.
The curve—
(118	)......y = Axm e™'
is solved by taking four ordinates, viz., for x, xk, xk2, xk3, when the
solution becomes1—
(119)	« - 1 log|log y2 ~2 lpg y* +lQgy4 i
( '......P ~ log k-Jog I log 2/1 - 2 log y2 + log y3)
using common logarithms. Then M denoting log e, we have also—
(1201 ^ (lQgy! - 21°gy2 + iog2/3)_(1ogy2 - 2iogy3 + iogy4)
1	Mx" {kP - 1) 2	MxPkP (kP - 1) 2
(121)	to = (lQg ^g ~ 1 og yi) ~ MnxrjkP- 1).
log Jc
There are obviously two other possible formula? for to.
(122).	.log A = log ?/i — m logxx + Mnx"
the value of M being 0.4342945. Three other formula"1 are also possible
for A. For further formulae see (150) to (153) later ; see also Figs. 21
to 27, hereinafter, for the forms of the curve.
8. Polymorphic and other fluctuations.—Monomorphic or rather
unimodal curves disclose a single maximum (or minimum) value. But
there are fluctuations which are polymorphic or multimodal. These may
be regarded as compounded of monomorphic curves. Practically their
dissection is best effected by the graphic methods of analysis. In general
any curve can be represented with great accuracy by either
(123	)......y = a + bx* + ex" + dxr + etc., or by
(124	)......Y= e " + bxr + 6x* + etc-
where p, q, r, etc., are not restricted to integral values.
The latter curve is reduced to the former by taking the logarithm ;
thus, y = log« Y. To solve for the constants we must have six points
besides the origin. If the value of a be known, the curve can be reduced
to one passing through the origin by subtracting a. Then we take values
of y for x, xk, xk2, xk3, etc. For the case for terms in p and q only,
we can proceed as follows :—
1 For a more complete study of the curve, see "Studies in Statistical Repre-
sentation. On the Nature of the Curve," above given, viz. (118), by G. H. Knibbs,
Journ. Roy. Soc., Vol. XLIV., pp. 341-367, 1910.