differences proceed irregularly in consequence ov a suficient number ov decimal places not having been taken to sho the law.

It may be remarkt that both Mr. Neison and Mr. Brown, after obtaining from their statistics data relating to each age, thro these together into quinquennial groops, and then make no further use ov the data at individual ages, but get from their groops values for quinquennial ages and simply interpolate for the other ages. It apears to me, however, that such a method ov procedure does not turn the observd facts to the best advantage. Insted ov taking the quinquennial values as fixt points ov departure, and simply interpolating, it is in my opinion better to uze som method ov ajustment-as, for instance, the grafic method-taking care to compare the ajusted with the original facts at each age. In this way we shal be abl to secure that our original facts ar not unduly alterd by the ajustment; but when we hav values only for quinquennial ages, we ar not so wel abl to juj how far it is alowabl to depart from our original values, and it is then safer to confine ourselvs to interpolation.

Mr. Woolhouse, in his paper abov referd to, describes a convenient practical method ov interpolation by means ov constant fourth differences, and remarks that hiher orders ov differences ar seldom employd, adding that perhaps in no case wil it be desirabl to proceed further than the sixth order ov differences. I find it convenient to employ constant fifth differences, for the folloing reason. Let six consecutiv quantitys, between which we hav to interpolate, be represented by the equidistant ordinats Pp, Qq, Rr, Ss, Tt, Uu, their values being yo, Y1, Y2, Y3, Y4, ys;

then, as abov explaind, the problem ov interpolating between y and Уз is the same as that ov drawing a curvd line between the points R and S, and, in order to get a satisfactory interpolation, it is necesary that this partial curv shoud join on smoothly to the ajacent partial curvs, namely, QR on the one side and ST on the other side. We shal secure this smoothnes ov junction, as regards QR and RS, if we so arange that these two curvs hav the same tangent and the same radius ov curvature at the point R; and

dy
dx

this wil be the case if the values ov and ar the same for

day
dx2

the two curvs at that point. Similarly, we may arange that at the point S the two curvs, RS, ST, shal hav the same tangent and the same radius ov curvature, or that the values ov dy, day

dx' dx2' shal be the same for the two curvs. The values of the differential coeficients at the two points R, S, hav to be determind by means ov the givn quantitys, and the simplest way ov determining them for the point R, wil be to supose a parabola ov the fourth order (or a quartic parabola) drawn thro the points P, Q, R, S, T; to determin the values ov the differential coeficients at the point R in this parabola, and then make the values for the two partial curvs QR and RS equal to these. Similarly, we may make the values at the point S in the two partial curvs, RS, ST, equal to those at the same point in the quartic parabola passing thro the points Q, R, S, T, U. The curv RS must therefore, in adition to passing thro the point R, satisfy five other conditions; namely, it must dy day pas thro the point S, and the values ov at the extremitys, dx' dx2' must hav givn values. We must therefore hav five disposabl constants in its equation. We may therefore asume it to be ov the form

Y2+x=Y2+ ax + bx2+ ca3 + dx1+ ex3

(1)

Since the curv is to pas thro the point S, it follos that when x=1, the value ov y wil be ys. This givs us the equation

It wil be convenient to expres the values ov the differential coeficients in terms ov the differences ov the givn quantitys. For this purpos, let them be differenst out in the usual way, as shoen in the folloing sceme :—

The equation to the curv ov constant fourth differences (or the quartic parabola) passing thro the points P, Q, R, S, T, is

·A2+

.

Differentiating this twice, we get

y=yo+xAyo+

1 X.X 1.2

B,

(6)

(7)

Now, let a, ẞ, 7, 8, be the central differences ov the five quantitys

a = {(Ay1+Ay2) = Ayo+&A2yo+{A3yo,

B=A2y1 = A2yo+▲3yo,

y= {(A3yo+A3y1)=A3yo+A1yo,
S=A+yo.

From these equations we find

Ayo=a-3B+y- - 92, A2= B−y+

2'

By substituting in (6) and (7), we get

2'

These ar the values ov the differential coeficients at the point R ov the quartic parabola passing thro the points P, Q, R, S, T.

dy2

The equation (1) shos us that for the partial curv RS,

=2b, and the condition that the differential coeficients shal