connected together by no law of continuity. If therefore the quantity sought after in any physical question, be given by the solution of a partial differential equation, this circumstance is itself a sufficient proof that the quantity is not subject to the law of continuity: it is a proof too that several quantities of the kind sought for may coexist. But of the infinite number of functions that will satisfy the differential equation, there will in general be a certain species which belongs in a peculiar manner to the question, and is to be determined by a discussion similar to that with which we commenced our consideration of the vibrations of an elastic fluid. No general rule can be given for such a discussion; the nature of the question itself must decide the manner of conducting it. It is essential that this primary form of the arbitrary functions be ascertained, before any application be made of the integral.

The views contained in this paper, I have found to be greatly confirmed by similar reasoning applied to the general equations of the motion of incompressible fluids.

TRINITY COLLEGE,

March 30, 1829.

J. CHALLIS.

XII. On the Comparison of various Tables of Annuities.

By J. W. LUBBOCK, Esq. B.A. F.R. & L.S.

OF TRINITY COLLEGE, CAMBRIDGE.

[Read March 30, 1829.]

1. IN last May I transmitted to the Philosophical Society of Cambridge some remarks upon the construction of Tables of Annuities; my object in that paper was to shew how the probabilities upon which Annuities depend, should be deduced from Tables of mortality, and I gave in illustration some Tables of Annuities calculated from observations of the mortality at Chester, by Dr. Haygarth, which appear to have been made with very great care. I have since compared these Tables with a great many others, and I now present the result of this comparison.

2. Very few registers of mortality give the deaths at every year throughout life, they generally give the deaths between birth and 5, 5 and 10, 10 and 20, 20 and 30 and so on for every decade. When the deaths are given between birth and 5, the living at 5, at 20, 30, &c. are known, and in order to form a complete Table of mortality it is necessary to interpolate the number of living at each intermediate age.

If the probability of an individual aged m years, living n years be called P, if r is the rate of interest, and if the same hypothesis of probability be adopted as in my former Paper which amounts to increasing by 1 the deaths at every age,

the value of a payment of unity after n years is

m being constant in this expression and n variable.

Instead however of interpolating values of Pm, n between those values which are known, it is better to interpolate at once between the values of Pm,n × (1+r)" which are given, but even this labour is unnecessary, because Σ annuity is a function of those terms only of the series which are given.

Let yo, Yi Yi, ••• Yni, Y(n+1)i, &c.

Pm,n
(1+r)n

be successive values of any variable y,

or the value of the

=

Yo + Yi + Y 2i + Yзi • • • • +Ym + Y (n+1); + &C. +Y (mni−1)

...

n(yo+Yni+Yeni........ + y)

+ i {1 + 2 + 3.

i

...

.n−1.} {Ay2+AYni + &c.+AY (m−1)ni}

+ {1.i−1 + 2.i−2+ &c. + n − 1.1} {A2y + A2yni + &c.},

1.2

1

1.2n2

==

A2ymni - A2yo

1

n

{1.n−1 +2.n-2 +3.n-3....+n-1.1} {Aym-^y.}

+

1

1.2.3 n3

{1.n−1.2n−1+2.n−2.2n−2....

+ n−1.1. n+1} {A2ym - Ayo}+&c.

m

The coefficient of A1y-A1y, is equal to the coefficient of x-1 in the development of

.1

or, in other words, if this coefficient be called %,

is the generating function of %, and since ni = 1,

Laplace has given in the 4th Volume of the Mécanique Céleste, p. 206, the particular value of this series which obtains when the interval i which separates the values of y is indefinitely diminished.

In this case the coefficient of Aym - Ay, is found by integrating

from i=0 to i=n, if n=1, the sum of the values of y or the area of the curve between ý, and ym

In applications of the former series to the calculation of annuities, reversionary payments, &c. ym, Aym, A2ym, &c. = 0.

The first term in the series of the values of y or y, is the value of a present payment = 1, if we neglect the term

and the following, and suppose the values of the annual payments to be in arithmetical progression, the value of an annuity on the life of a person aged 20, to commence at the end of the first year.