66

to distance from the centre. But the only admissible law of distribution as to distance from the centre is the law of uniformity; they must not be closer towards the circumference than they are at the centre. If we suppose the chords drawn so that they are distributed evenly as to direction, and also evenly as to distance from the centre, the probability of two chords intersecting is certainly. This is the solution of problem (B), and it is in accordance with Professor CROFTON'S system or definition of random lines. On this principle, I had given a Solution of Question 2109 (Reprint, Vol. XI., p. 94), before Professor CROFTON had given his theory of random lines, which Mr. WOOLHOUSE characterizes as peculiar. But Mr. WOOLHOUSE himself had adopted this system, and extended it to space, in solving the following problem of his own in the Diary for 1860:"A cube is thrown into the air at random, and a shot fired through it, what is the chance the shot passes through opposite faces of the cube " To get the result, Mr. WoOLHOUSE determines for any one definite direction, (1) the number of shot (or parallel lines) which, evenly distributed, would pass through the cube supposed fixed: (2) the number-being part of the preceding-which pass through two opposite faces; (3) embracing all directions evenly, he determines the total number of lines which intersect the cube; and (4) the total number which pass through two opposite faces; the ratio of these last numbers being the required result. In the same number of the Diary, another correspondent solves the question by supposing the shot to enter one particular face, and from that entrance point-the cube being fixed-to take any direction at random. This solution Mr. WOOLHOUSE shews to be erroneous. We have in this case random lines through random points in a face of the cube taken in place of random lines. And on the same principle I object to the solution that gives as the answer to Question 5461, because in that solution lines drawn through random points in a circle are taken for random lines. But we must not confound random lines through random points on a limited space, with random lines wholly unrestricted.

2. On sending the foregoing remarks to Professor CROFTON, we received from him the following note thereon:—

Of course ས་ a random

I quite agree with Col. CLARKE's remarks on Quest. 5461. the results will depend on the manner in which we suppose chord " of the area to be drawn. The results obtained by Miss BLACKWOOD and others are quite correct on the supposition that two points are taken on the perimeter of the circle (or any other convex figure) at random, and joined. I would observe, however, that it would be just as natural to take any two points at random within the circle and join them; but the result would be quite different; the problem indeed would, in this form, be a very difficult one, and would be an interesting exercise for our contributors. My own idea of the most natural sense of the term is a straight line drawn at random in the plane, and meeting the area-that is, the plane is supposed covered by an infinite number of random lines, and those which cross the area alone are considered; one of them being taken at random: exactly as a random point might be supposed to be selected from those, out of an infinite number scattered over the plane, which happen to fall inside the boundary in question. Thus we might imagine an infinity of points to fall like drops of rain all over the plane, and only consider those that fall on the figure: and, in the former case, an infinity of lines thrown down at random on the plane in the same way. Of course it comes to the same thing to suppose a single point thrown at random an infinite number of times on the plane; or a single straight line; and its position marked out every time it falls on the boundary. This coincides with M. BERTRAND's view,

which is expressed as follows on p. 487 of his Calcul Intégral :-" Concevons sur un plan une série de lignes parallèles équidistantes, et supposons que l'on projette au hasard sur ce plan un disque de forme convexe de dimensions assez petites pour ne pouvoir pas rencontrer deux lignes à la fois et qu'on recommence l'épreuve jusqu'à ce qu'il y ait intersection. La corde détachée par celle des lignes qui est rencontrée sera pour nous la première des cordes choisies au hasard; la seconde s'obtiendra par une autre épreuve, renouvelée, s'il est nécessaire."

3. Mr. G. S. CARR, referring more especially to the form that Mr. WOOLHOUSE gave to the question, writes as follows:

If two persons enter a dark room and draw independently a line with a ruler upon a circular slate, it appears that what would happen would be this:-Each person would approach the circle indifferently from any point of the compass, and, holding the ruler before him in the usual manner, would place it upon the circle at any indifferent distance from himself. Consequently, the two random lines so drawn would fall under Professor CROFTON'S definition of random lines, and the probability of their intersecting would certainly be. It would seem that a line drawn through one or two points, even though those points be taken at random on a given line or area, is to be considered "at random" only in a restricted sense. But the definition founded upon the two elements of angular direction and translation alone, is absolute and covers all cases.

4. Miss BLACKWOOD, after reading a "proof" of the foregoing part of these Notes, calls the EDITOR's attention to the fact that Mr. WOOLHOUSE has "pronounced her Solution of Question 5461 to be clear and satisfactory, and needing no confirmation," and then, remarking that she has thus "drawn upon herself the heavy artillery of Colonel CLARKE and his allies," replies thereto as follows:

It is positively refreshing to find Mathematicians now and then throwing aside their ponderous mathematical armour, and indulging in a pleasant set-to on the neutral ground of metaphysics. At the same time, it seems to me that the practised logician has here the same unfair advantage over the mathematician that FITZ-JAMES had over the chivalrous but ill-fated RODERICK DHU at Coilantogle Ford, when the latter threw down his "targe," and trusted to his sword alone.

The question at issue is the purely metaphysical one, what should be meant by the phrase "a random chord." The logician (FITZ-JAMES) says to the mathematician (RODERICK DHU), "Let us strike out the word random for a moment, and consider what is the ordinary accepted definition of the simple word chord." Referring to a mathematical text-book, they find and accept the definition, “A straight line joining two points in the circumference of a circle is called a chord." "What two points in the circumference ?" asks the logician. Any two points you like to take," answers the mathematician; and, so saying, he exposes himself to the decisive thrust, "Then, when the points are random points, the chord must, as a necessary consequence, be a random chord."

66

5. Commenting on a "proof" sent to him of Nos. (1), (2), (3) of the foregoing notes, Mr. WOOLHOUSE writes as follows:

In these Notes on Random Chords no allusion whatever is made to the principal matter, viz., the errors pointed out, upon which an exercise of judgment might have been of some service. The notes, indeed, are merely responses to a passing observation, which must be under

stood as having exclusive reference to the subject of "random chords drawn in a circle," that Professor CROFTON's theory when applied to such chords is peculiar, and that results derived from it, should be regarded as special. My meaning is more fully stated in the Note on Random Lines (Reprint, Vol. X., p. 33). It would seem, however, that Colonel CLARKE had written his remarks without giving much, if any, consideration to that Note, although it is expressly referred to by me. The only divergence of opinion does not relate to any process of investigation, but simply to the definition given to "random chords" at the outset, which to a certain extent may be considered as optional, even at the risk of being peculiar, provided the same be clearly explained. I have, however, before stated, and am prepared to maintain, that the mathematical condition to which random lines are subject, when not specially defined, is not arbitrary, but should in every case be elicited from the practical nature of the problem, as suggestive of the particular mode in which the lines may happen to be generated. The cube and circular-slate questions are both accurately solved on this undoubtedly rational and correct principle. With respect to the modus operandi of drawing the chord across the circular slate, I object to the arbitrary description given by Mr. CARR. I hold that each person would, in the first instance, instinctively ascertain a locality in the periphery from whence to draw the chord, and that the chord so drawn would therefore be a random line proceeding from a random point in the circumference, and would not fall under Professor CROFTON's general definition of random lines. In short, I consider Professor CROFTON's definition as not adapted to ordinary notions of the generation of a random chord; but at the same time there can be no doubt whatever, that it is the best definition of a "random line on an indefinite plane." Adapting the alternative enunciation to the case before us, I am therefore of opinion (1) that the probability of intersection of "two random chords drawn in a circle," is ; and (2) that, if "two random lines meet a circle" the probability that they shall intersect within the circle is. If any other mathematicians perceive sufficient reason to adopt the latter as "random chords," it is no business of mine.

4870. (By Professor CAYLEY, F.R.S.) Given three conics passing through the same four points; and on the first a point A, on the second a point B, and on the third a point C. It is required to find on the first a point A', on the second a point B', and on the third a point C', such that the intersections of the lines

A'B' and AC, A'C' and AB, lie on the first conic;
B'C' and BA, B'A' and BC, lie on the second conic;
C'A' and CB, C'B' and CA, lie on the third conic.

Solution by J. HAMMOND, M.A.; Prof. EVANS, M.A.; and others. Taking ABC as the triangle of reference, the three conics are uby2+cz2 + fyz + gzx + hxy 0, v = a2x2 + c'z2 + f'yz + g'zx + h'xy = 0, w= a'x2+b" y2+ƒ"yz+g"zx + h′′xy = 0,

=

subject to the condition wu+λv, since they pass through the same four points. Now B'C' passes through the intersections of BA and v and of CA and w, that is, B'C' passes through the points

=

0),

(a'x+h'y = 0, z = 0), (a"x+g'z = 0, y its equation is therefore Pa'a′′x+a"h'y+a'g′′z = 0. Similarly, the equations of C'A' and A'B' are Q=b"hx+b"by+bf"z = = 0, Rcgx+cf'y + cc'z = 0.

Also, since wu+λv, we have ƒ" = ƒ +λƒ', b′′

5474. (By the Rev. A. F. TORRY, M.A.)-Find what normal divides an ellipse most unequally.

Solution by H. POLLEXFEN, B.A.; J. L. MCKENZIE, B.A.; and others.

Let POQ be the required normal, P'OQ' the consecutive normal. Then the difference between the areas PAQ and

PA'Q, being a maximum, is for the moment constant; and the triangles POP', QOQ' are therefore equal. Hence POQ is bisected in O; or the chord PQ is the diameter of the circle of curvature at P. But the tangent at any point of an ellipse, and the common chord of the ellipse and its circle of curvature, make equal angles with the axis. Therefore, in the present case, the tangent and normal at P make angles of 45° with the axis. The coordinates of P are

x = a2 (a2 + b2), y = l2 (a2 + b2)−1.

QQ

5428. (By Professor ELLIOTT, M.A.)-Prove (1) that the highest point on the wheel of a carriage rolling on a horizontal plane moves twice as fast as each of two points in the rim whose distance from the ground is half the radius of the wheel; and (2) find the rate at which the carriage is travelling when the dirt thrown from the rim of the wheel to the greatest height attains a given level, explaining the two roots of the resulting equation.

Solution by J. J. WALKER, M.A.; Prof. EVANS, M.A.; and others.

The first part of the question is obvious, since the velocities of all points on the wheel are proportional to the chords drawn from the point of contact with the ground to those points.

=

For the second part it is readily found that, V being the velocity of the entre and V2 2gh, the height on the wheel from the ground of the par4.2 ticle of mud which reaches the greatest altitude is r+

2h

which being

necessarily not greater than 2r, 2h must be not less than r. And supposing a to be the greatest height attained, 2h is determined by the equation

4h2-4 (ar) h + r2 = 0,

so that one of the two values of 2h is greater and the other less than r. The latter value is therefore excluded by the nature of the Question.

5437. (By CHRISTINE LADD.)—If I, I, I, be the points of contact of the inscribed circle with the sides of a triangle ABC; 01, 02, 03 the centres of the escribed circles; ri, re the radii of the circles inscribed in the triangles III, О100 ̧; and a, B, y the distances 0203, 0301, 0102; prove that

ri

re

=

2R

a + b + c a+B+7

Putting a, b, C1, 81

III, we have a

Solution by the EDITOR.

for the sides and semiperimeter of the triangle
2r cos A, bi 2r cos B,
2r cos C; and as

ri, r are the respective radii of the inscribed and circumscribing circles of this triangle, we have, by a well-known form,

New

a+b+c a+B+ y2 Geometrical

the last form following from Vol. II. of Dr. BooтH's Methods, Secs. 187, 216, where it is proved that

=

4R cos A cos B cos C
Again, the angles of the triangles III, 010203 are

s, 4R (cos A+ cos B+ cos C) = a + B + y.

(B+C), (C+A), (A+B);

hence these triangles are similar; and as the triangle ABC is the orthocentric triangle of the triangle 0,0203, 2R is the radius of the circle that circumscribes the triangle 0,0203;