7 ss PROBABILIT Y XA, XB to the inner. Find the mean value of the arc AB. We shall find M(AB)-LS/A, L being the whole length of the inner curve ABV. We will first prove the following lemma : If there be any convex arc AB (tig. 13), and if Xj be (the measure of) the number of random lines which -^^ meet it once, N._> the number which meet ^/^ ^, it twice, A B 2arcAB = X, + 2X,. Fig. 13. For draw the chord AB ; the number of lines meeting the convex figure so formed is Xj -f X., = arc -f chord (the perimeter) ; but Xj = number of lines meeting the chord = 2 chord; . . 2 arc + Xj = 2Xj + 2X., , . . 2 arc = N x -f 2X 2 . Xow fix the point X, and draw XA, XB. If a random line cross the boundary L, and p l be the probability that it meets the arc AB once, ;> 2 that it does so twice, and if the point X range all over the an- uulus, and p v p.^ are the same probabilities for all positions of X, 2M(AB)/L-. Pl + 2p J . Let now IK (fig. 14) be any position of the random line ; drawing tangents at 1, K, it is easy to see that it will cut the arc AB twice when X is in the space marked a, and once when X is in either space marked tion of the line, Fig. 14.
- hence, for this posi-
P! -f 2/? 2 = I 2S .... = -r- , which is A hence M(AB) S Hence the mean value of the arc is the same fraction of the perimeter that the constant area S is of the annulus. If L be not related as above to the outer boundary, M(AB)/L = M(S)/A, M(S) being the mean area of the segment cut off by a tangent at a random point on the perimeter L. The above result may be expressed as an integral. If s be the arc AB included by tangents from any point (x, y} on the annulus, ffsdxdy = LS. It has been shown (Phil. Trans., 1868, p. 191) that, if d be the angle between the tangents XA, XB, The mean value of the tangent XA or XB may be shown to be where P = perimeter of locus of centre of gravity of the seg ment S. 82. If C be the length of a chord crossing any convex area n ; 2, 2 the areas of the two segments into which it divides the area; and p, u the coordinates of C, viz., the perpendicular on C from any fixed pole, and the angle made by p with any fixed axis ; then ff&djKlu = 6 jfcs djMlu, both integrations extending to all possible values of p, &> which give a line meeting the area. This identity will follow by proving that, if p be the distance between two points taken at random in the area, the mean value of p will be MO>) = o -2/722 . M, ...... (i), and also M(p) = JQ-^C 4 <(pd ...... (2). The first follows by considering that, if a random line crosses the area, the chance of its passing between the two points is 2L- 1 M(p), L being the perimeter of ft. Again, for any given position of the random line C, the chance of the two points lying on opposite sides of it is 222 H 2 ; therefore, for all positions of C, the chance is 2n- 2 M(22 ); but the mean value M(22 ), for all positions of the chord, is M , ffitidpii* tr -- = ffdpd* To prove equation (2), we remark that the mean value of p is found by supposing each of the points A, B to occupy in succession every pos sible position in the area, and dividing the sum of their distances in each case by the whole number of cases, the mea sure of which number is fl 2 . Con fin- T ing our attention to the cases in which the inclination of the distance AB to some fixed direction lies between 6 and 6 + dO, let the position of A be fixed (fig. 15), and draw through it a chord HH = C, at the inclination ; the sum of the cases found by giving B all its positions is /* /" ./o _/o where r = AH, ?- = AH . Now let A occupy successively all posi tions between HH and kk , a chord parallel to it at a distance = dp ; the sum of all the cases so given i be Xow, if A moves over the whole area, the sum of the cases will be where jo = perpendicular on C from any fixed pole 0, and the integration extends to all parallel positions of C between two tan gents T, T to the boundary, the inclination of which is 0. Remov ing now the restriction as to the direction of the distance AB, and giving it all values from to IT, the sum of all the cases is or, if o> = inclination of p, du> = d6, and the sum is The mean value of the reciprocal of the distance AB of two points taken at random in a convex area is easily shown to be Thus, for a circle, 16 It may also be shown that the mean area of the triangle formed by taking three points A, B, C within any convex area is M(ABC) = n - Ci-*ff($-S?dpdu . 83. In the last question if we had sought for the mean value of the chord HH or C, which joins A and B, the sum of the cases when A is fixed and the inclination lies between and 6 + dd would have been and when A lies between II H and hh Cdedp/ V 2 + r -)dr and finally, the mean value of C is Thus the mean value of a chord, passing through two points taken at random within any convex boundary, is double the mean dis tance of the points. 84. } T e have now done enough to give the reader some idea of the subject of local probability. We refer him for fuller information to the very interesting work just published by Emanuel Czuber of Prague, Geometrische WahrschcinlicJikeiten und Mittclwcrtc, Leip- sic, 1884 ; also to the Educational Times Journal, in which most of the recent theorems on the subject have first appeared in the form of questions, under the able editorship of Mr Miller, who has him self largely contributed. In AVilliamson s Integral Calculus, and a paper by Prof. Crofton, Phil. Trans., 1868, the subject is also treated. Literature. Besides the works named in the course of this article, see De Morgan s treatise in the Encyclopedia Metropolitans; Laurent, Tratte au Calcul des Probabilities, Paris, 1873 ; Gourand, llistoire du Calcul des Prob., Paris, 1848; J. V. L. Glaisher, "On the Law of Facility of Errors of Observations, and the Method of Least Squares," Trans. K.A.S., vol. xxxix. ; Cournot, Theorie des Chances; Liagre, Calcul des Prob.; General Didion, Calcul des Prob. applique au tir des projectiles. Those who are interested in the metaphysical aspect of the question may consult lioole s Laws of Thought, also J. S. Mill s Logic. To these and the other works we have named we refer the reader for an account of what we have had to omit, but above all, to the great work of Laplace, of which it is sufficient to say that it is worthy of the genius of its author the Theorie analytique des Probability s. It is no light task to master the methods and the reasonings there employed; but it is, and will long continue to be, one thnt must be attempted by all who desire to understand and to apply the theory of
probability. (M. W. C.)