The assumption expressed by

ƒ (x2) ƒ (y2) = ∞ (xy)

is therefore either a simple mistake or a petitio principii: the former, if it is deduced from the general principle that the probability of a compound event is equal to the product of those of its elements; the latter, if it is made to depend on the particular form assigned to f(x).

After all, too, if the demonstration were right instead of wrong, it would not prove what is wanted. For if the law of probability of a deviation parallel to a fixed axis is expressed by the function

h

e-h2x2 dx,

which is what the amended demonstration tends to show, the probability that the stone falls on the area dxdy is plainly

Transforming this to polar co-ordinates, and integrating from 0 to 27 for the angle vector, we get 2h3e-h22 rdr for the probability that the deviation from the mark lies between r and r+dr; a result which may be verified by integrating for r from zero to infinity, the integral between these limits being equal to unity. Thus if the deviations measured parallel to fixed axes follow the law which the reviewer supposes to be universally true, the deviations from the centre or origin follow quite another; and hence it appears that his illustration is altogether wrong. For if 2he-h2r rdr is the probability of an error lying between r and r+dr, the centre of gravity of the shot-marks is not the most probable position of the wafer. So that his hypothesis is self-contradictory.

The original source of his error was probably the analogy between Gauss's law, and the limiting function in Laplace's investigation.

I am, my dear Sir,

BRIGHTON, Sept. 19.

Most truly yours,

R. L. ELLIS.

NOTE TO A FORMER PAPER ON AN ALLEGED

PROOF OF THE METHOD OF LEAST

SQUARES*

To the Editors of the Philosophical Magazine and Journal.

GENTLEMEN,

ALLOW me to correct an error in my letter to Professor Forbes, published in your last Number. The Edinburgh reviewer, on whose proof of the method of least squares I was commenting, says that the most probable position of the wafer is the centre of gravity of the shot-marks; of course on the supposition that in this, as in all other cases, the probability of a deviation or error r is equal or proportional to a certain constant base raised to the power-2.

Now, admitting this supposition to be true, the centre of gravity is not the most probable position of the wafer. But, on the contrary, if the function mentioned at the close of my former communication, viz. 2h'e-hr rdr, expresses the probability of an error r, then the centre of gravity is the most probable position. I thus not only omitted to notice that the reviewer's conclusion would not follow from his own hypothesis, but by this omission was led to introduce an error of my own.

It is unnecessary to trouble you with a proof of what I have now said, as the matter does not affect the general question.

BRIGHTON, Nov. 7.

I am, Gentlemen,

Your obedient Servant,

R. L. ELLIS.

* Philosophical Magazine, December, 1850.

ON SOME PROPERTIES OF THE PARA

BOLA*.

THERE are many very interesting properties of the Conic Sections which are not to be found in the usual works on the subject, but are scattered through various memoirs in scientific Journals. Those relating to the properties of polygons inscribed in and circumscribed round conic sections, have been investigated by a great many writers both in France and England. Pascal was the first who engaged in these researches, and was led by the curious properties which he discovered to call one of these polygons the "hexagramme mystique." After him Maclaurin gave a proof of a theorem which is not only beautiful in itself, but also very fertile in its consequences. In more recent times Brianchon has demonstrated the remarkable theorems, that in all hexagons either inscribed in or circumscribed round a conic section, the three diagonals joining opposite angles will intersect in one point. Subsequently, Davies in this country, and Dandelin in Belgium, proved in different ways the same propositions along with others. The latter adopted a very peculiar method, deducing these and many other properties of sections of the cone by considering the cone as a particular case of the "hyperboloide gauche." Generally speaking, the Geometrical method is more easily applied than the Analytical to these cases, and accordingly all the proofs given have depended on geometry, with the exception of one published by Mr Lubbock in the Number of the Philosophical Magazine for August 1838. He has there demonstrated, by analysis, Brianchon's Theorem for a circumscribing hexagon in the particular case where the conic section is a parabola; but his method is tedious, and not remarkable for symmetry and elegance, so that another proof is still desirable. The following one is founded on the

* Cambridge Mathematical Journal, No. V. Vol. I. p 204, February, 1839.

form of the equation to the tangent of the parabola which is given in Art. 2 of our first Number*.

Let the parabola be referred to its vertex, then the equation to its tangent by that article is

where a is the tangent of the angle which the tangent makes with the axis of y. If a be the corresponding quantity for another tangent, its equation will be

Combining these equations, we shall find for the co-ordinates of the point of intersection of the two tangents

x = max', y=m (a + a').

We shall distinguish the tangents which form the different sides of the hexagon by suffixing numbers to the a which determines their position, and we shall likewise distinguish the coordinates of the summits of the hexagon by suffix letters.

The equations to the three diagonals are these:

(1) y (αα-αα) — x (α ̧ + α ̧—α ̧ — α2)

4 5

4

Expressions which, as they ought to be, are symmetrical with respect to the a's.

Multiply (1) by α, (2) by — α, (3) by α, and add. Then y will disappear, and we shall find

Again, multiply (1) by α, (2) by -a,, (3) by a,, and add: as before, y will disappear, and we shall find the same value for x. Consequently two straight lines whose equations are

Cambridge Mathematical Journal, Art. 2, No. I. Vol. 1. p. 9.

and which have a point in common, cut (3) in points whose abscissæ are equal, and which therefore coincide. Hence either two straight lines enclose a space, or (3) passes through the intersection of (1) and (2). Thus the existence of the point common to the three diagonals has been proved, and its abscissa found. To determine its ordinate, add (1), (2), (3), when x disappears, and we have

If we call the co-ordinates of the point where the third and sixth sides of the hexagon meet xy, and so of the other two points, these expressions for x and y become

x =

У

Y

These expressions, as of course we should expect, are symmetrical.

In the last Number of this Journal a demonstration was given of a property of a parabola: That the circle which passes through the intersections of three tangents also passes through the focus. Although six demonstrations of this theorem have already appeared, yet the following is so simple that its insertion here may not be inappropriate.

Referring the parabola to the focus as origin, we can put the equation to the tangent under the form

where a is one-fourth of the parameter, and m the trigonometrical tangent of the angle which the tangent makes with the axis of y. Hence if x, y, be the co-ordinates of the point of intersection of