STUDIES IN STATISTICAL REPRESENTATION. 489 
13. Curves which are the products of parabolic and 
exponential curves.—The application and solution of a curve 
which is the product of the parabolic or hyperbolic and 
exponential curve, is dealt with in a paper entitled ‘* On 
the Nature of the Curve 
OS Lie ES aire (38) } 
14. On the curve which is the sum of a series of para- 
bolas, or of a series of hyperbolas, or both.—Consider the 
curve y = a + bx?+ cxv4+ da™+ etc....in which p,q, r, etc., 
may be fractional. Since each term has greater fitting 
power than when restricted to an integral value of the 
index, it is obvious that the sum of several terms has also 
greater fitting power. Within the region of possibility if 
there be m indices, the curve will pass through 2n + 1 
points, whereas if the indices are integers the curve will 
pass through n + 1 points only. 
It will be well to consider first the case where there are 
two terms in x only, that is, two indices p and q of which, 
let us suppose, q is the greater. By taking the origin on 
the curve the equation assumes the simpler form 
7 OTD a A er AA (39) 
and we shall primarily consider this case, viz., where a=0. 
The determination of the constants may then be effected 
as follows:—Taking, as before, four ordinates whose 
abscisse are in geometrical progression, viz., x, xk, xk’, xk’, 
we have 
= UL? cx 6 Op == Cn?) BP AN | 
ae oe” ce ye be. bee + pee 
For $<? write L; and for cx* write M: for k? write «; and 
for k* write 6; the four preceding equations (40) are then 
equivalent to 
y=L+M; y,=La+MUB; y,= Lo? + MB’; y,= Lui + MB?*...(41) 
+ G. H. Knibbs, c.u.c., etc. Journ. Roy. Soc. N.S.W., Vol. xxiv, pp: 
341-367. 
