Practical Astronomy in the University of Edin. (Edin. 1818). In the University Library there is a second edition of a part of the same work with the title Scientific Aphorisms (Edin. 1827). I bring them before the notice of the Society, as they contain an explanation of gravitation, &c., almost identical with that of Le Sage, to which our attention was lately recalled by our President. Professor Blair seems to have invented this explanation for himself-because, though he gives frequent references to other authors, whose results he quotes, he makes, so far as I have seen, no reference to Le Sage. On a future occasion I may enter on a discussion of the points of resemblance and difference of these two theories.

5. Note on an Identity. By Professor Tait.

Whatever be p and 2 it is obvious that

absolutely without any restriction on the values of the quantities involved.

It is obvious that an immense number of curious results in the form of sums of series, &c. can be derived with great ease from this expression and from various modifications of it. I give, therefore, only a few very simple examples.

Take 1, 2, &c., as the first n of the natural numbers, and the series becomes

whence at once the sum of the first n + 1 terms of the expansion of (1-1) is seen to be

We obtain merely the same result if we take 91, 92, &c., as any set of consecutive whole numbers; but from the theorem itself it is easy to obtain the equality,

If in this we write each letter for its reciprocal we have

p = q1 + 1/2 (p − q) + 2/2 (p−91) (p −q) + &c.,

p2

1=

= cos 0+ cos 20(1 − cos 0) + cos 30(1 - cos 0)(1 − cos 20) + ...

+ cos no(1 - cos )...(1 - cos (n - 1)0)

+ (1 - cos 0)(1 − cos 20)...(1 - cos no),

of which a very interesting case is given by no 2π.

=

As a final example we have the singular for mula,

whence it follows that, subject to the introduction of the remainders as above (which vanish if the series are extended to infinity,

By another application of the formula we may easily obtain finite expressions for the sum of the series of which two successive terms

I obtained the first expression above by integrating by parts a power such as P-1, but the following mode of obtaining it shows at once its nature.

Let there be a number of independent events, A, B, . . . . N, whose separate probabilities are a, ẞ, ... v. Then the chance that one at least of them occurs is

1-(1-a)(1-8).... (1-v).

But we may obtain another expression for the same result by writing the chance that any one (say A) occurs, adding to that the chance that another (say B) occurs while A does not occur, then that C occurs and neither A nor B, &c. This gives

Equating these two expressions we get an identity which is easily transformed into that first given.

But its truth is much more easily seen if we write a' for (1 − a), &c., when the last given form becomes

1-a'ẞ'y....1- a' + (1 − ẞ')a' + (1 − y')a'ß' + . . . .

=

which is an obvious truism. The method seems well worth the attention of any one with leisure and some analytical skill.

July 24.-Mr Muir has kindly given me a reference to Crelle, vol. xii. p. 354, where it is stated that the above identity in one of its forms is in Schweins' " Analyse," p. 237. This work I have not seen. Mr Muir adds that no developments or applications of the theorem are made.

The following Gentlemen were duly elected Fellows of the Society:

1. JAMES STEVENSON, 4 Woodside Crescent, Glasgow.

2. JAMES ROBERTON, LL.D., Professor of Conveyancing, 1 Park Terrace East, Glasgow.

3. GEORGE A. PANTON, 24 Bennet's Hill, Birmingham.

4. ISAAC BAYLEY BALFOUR, Sc.D., 27 Inverleith Row.

5. Sir DANIEL MACNEE, 6 Learmonth Terrace.

6. WILLIAM POLE, Mus.Doc., Memb. Inst. Civil Engineers, 31 Parliament Street, Westminster.