I think that no reliance is to be placed on the places for the 2nd and 4th of October. It has already been noticed how difficult and uncertain the observations of the latter day were: those of the former present an anomaly for which I am at a loss to account. Though they appear to agree with each other, they indicate a retrograde instead of a direct motion in right ascension. The cause of this error I cannot assign. Excluding these two places, I have compared the others with Mr Stratford's Ephemeris in the N.A. for 1839, in which the perturbations are taken into account. I have obtained the following results for the differences in right ascension and declination; where it will be observed that indicates that the observed right ascension or declination exceeds that found by calculation, and also that the differences in right ascension have been reduced from the equator to the parallel of the Comet.

entirely, though of course they They are, in several instances, This is the case in 19th October. This It rests upon five

These differences do not depend do partly, on errors of observation. largest on the places that are best determined. a remarkable manner with the declination of the is the best determined place of the whole series. observations in good accordance with each other; and the star of comparison is in Mr Pond's last Catalogue, and consequently is known with great accuracy.

LONDON, Nov. 23, 1837.

R. W. ROTHMAN.

XXVI.

Mathematical Investigation of the Effect of Machinery on the
Wealth of a Community in which it is employed, and on the
Fund for the Payment of Wages. By JOHN TOZER, ESQ. B.A.
of Caius College.

[Read May 14, 1838.]

IN the third and fourth Volumes of the Society's Transactions are two papers by Mr Whewell, in which symbolical language is applied to the solution of some problems of Political Economy. In the following Paper another problem of the same science is subjected to a similar mode of investigation.

An opinion has been expressed that the term Political Economy has acquired an extent and a vagueness of meaning which in a high degree unfits it for the purposes of science. Certainly any attempt to apply mathematical reasoning to all the subjects which have by different writers, and at different times, been included in the name, must be altogether unsuccessful. Neither do we possess the data, nor has analysis the powers, necessary to such a task.

If, however, the investigation of the causes which affect the accumulation and distribution of wealth, be kept distinct from any considerations as to the effect of that accumulation, or the mode of its distribution on the happiness of mankind, and be also separated from any speculations or deductions as to the nature of those political and social institutions by which these causes may be modified or brought ♦ into action; the science that results, by whatever name it may be called, acquires an almost entirely demonstrative character-becomes a

series of propositions which are logical deductions from assumed definitions, and form those properties of the things defined which furnish axiomatic truths, and is therefore a subject to which mathematical reasoning is not only proper but necessary.

The introduction of any process by which accuracy may be given to the reasonings of Political Economists, as tending, however remotely and indirectly, to place the science in this position, must be valuable; and even if we could suppose the principles to be as accurate in their enunciation, and as complete in their demonstrations as they can be rendered, there would still devolve on us the duty of rendering our deductions from them general, and of proving that they were necessary.

The particular problem under consideration, is of very limited extent, and of very easy solution. The method that has generally been employed has been to take particular numerical examples, and the results of these have frequently been assumed to lead legitimately to general conclusions. If the examples chosen had always been supplied by statistical facts, we should at least have been assured that the phenomena displayed in these results either had or might have happened, however unsatisfactory the general conclusions from them might have been.

This advantage, however, has not been afforded, the numbers have been generally assumed without reference to realities, and though it may sometimes have been carefully stated, that the conclusions could not possess a higher degree of truth than the premises, the impressions on the minds of general readers would be favourable to that particular conclusion which the example chosen tended to support.

PROP. A portion of capital, which either has been or would have been employed in the payment of wages, is used in the construction of machinery; to determine the effect on the wealth of the community, and on the fund for the payment of the labourer.

Let C be the capital,

q-1 the ordinary rate of profit.

If C be engaged in paying wages, it will yield annually a profit = C(q-1); this, while the machinery is in progress, must be partly taken from capital.

Suppose the machinery to take b years to complete, and a part

C

of C = to be expended every year in its construction till it is com

pleted, then the productive capital will be diminished,

and at this time the whole capital will have been expended on the machine;

Let a fractional part, the m" of the expenditure for machinery, go to the labour, and B1, B...... B be the sums by which the fund for paying wages is diminished during the 1st, 2nd progress of the machinery.

Thus

......

pth years of the