(22)

bn(p + 1)

m + pn

+

an(p + 1)

m + pn.

m + pn + n

an(p + 1)" Sxm-1dxX P+1. It should also be observed that equations (14), (15) and (16) may be obtained directly by following the general method by which (1) and (2) were derived, and this method will be found much more simple than that generally pursued in obtaining reduction formulae for binomials.* *Erratum. On page 138, line 6, for "general reductions " read general relations.

USEFUL FORMULAE IN THE CALCULUS OF FINITE DIFFERENCES.

BY G. W. HILL.

The finding of the values of the differential coefficients of a function of a single variable and of the single and double integrals with respect to the independent variable, from special values of the function computed at equidistant intervals, is an operation very frequent in Planetary Astronomy. The following seems a simpler exposition of the matter than has hitherto been given :

Let y be a function of a computed for the series of values of x, a h, a, ah, a + 2h, ....; and let the differences and first

and second summed values of y be denoted thus,

With regard to the differences of odd orders let us adopt the general notation

Jantry; = } { jan+ly, -; + £ently1+4),

i

i+

n and i being integers. In this way the symbol does not follow the law of indices as in the ordinary method of differences, that is we do not have in general

Nevertheless it is evident the following relations hold :

jenjan' = £2(n+n'), 12n+1 J2 =

that is, the exponents are to be added except when both are odd.

d dx'

For brevity writing D for and e denoting the base of hyperbolic

logarithms, the symbolical expression for Taylor's Theorem gives

the integral being taken so as to vanish with J. The operation denoted

by the symbolical expressionis evidently a function of 4, and we

have

It is plain, then, in general, that we have the value of an even differential coefficient from the formula

Differentiating the first of these with respect to 4, we obtain

Thus the value of an even differential coefficient can be obtained from that of the preceding differential coefficient by the very simple formula D2" = 2nh-Dɔn-1dd.

If we employed the preceding formula for D for expanding its value in powers of 4, we should find it difficult to discover the law of the numerical coefficients, but by differentiating the value of D2n+1 with respect to 4, we shall find that it satisfies the differential equation

we shall find that the coefficients A, satisfy in general the relation

And as we know that A, =1, this suffices for obtaining all the coefficients in succession. In the general case if we put

D2nh-2nY.A(2), D2n+1h-2-1. A(2n+1),

si,

the above differential equation gives the following relation between the coefficients :

by which all the coefficients in succession may be derived.

where the law of the coefficients is readily seen.

In the higher differ

ential coefficients the fractions are more complex, we therefore content ourselves with writing the values thus,

The above expressions for D2

and D2+1 are equally applicable when

n is negative; they then give the formulae to be used in mechanical quadratures, thus

If these expressions are expanded in powers of ▲, we obtain

+

317 22809600

A8

..).

1

These are the expressions to be used in computing the values of the integrals Sydx and Sydx. It must be noticed that virtually contains an arbitrary constant C, and 4-2 an arbitrary expression Cx + C'. In fact the quantities in the columns to the left of that of the function y cannot be written until we know one quantity in each column. These constants C and C' are usually determined from the given values of fydx and ffydx2 for xa. If we denote them by D1 and D-2 and if in general the subscript (,) denote values which obtain when x= a, it will be seen that

Having thus the sum and difference of the quantities ▲ly_ and

1y, it will be easy to get the quantities themselves.

14

In using the method of mechanical quadratures it is usual to multiply the values of y by h, if the single integral only is wanted, but by h2 if the double is also to be obtained; in the last case then it is necessary to divide the results obtained by h in order to have the single integral.

These formulae appear to have been first obtained by Gauss (Werke, Vol. III, p. 328). Encke has given them in the Berlin Jahrbuch for 1838. For use they are much superior to the formula given by Laplace (Mecanique Celeste, Vol. IV, p. 207).

PROBLEM IN "CURVES OF PURSUIT," BY WERNER STILLE, MARINE, ILL.-A hunter is in the field with his dog. The hunter, standing at a point A calls his dog, which is at a point B. At this moment the hunter walks off in a direction right-angular to AB, while the dog, running n times as fast as his master walks, keeps his master continually in view, thus at each moment running along the right line which then lies between him and his master.

Find thn Curve along which the dog runs until he overtakes his master at C.