170 Mr. J. M. Wilson on some Problems in Chances. 



BischofF (Quart. Journ. of Science, October 1864, p. 688) in a 

 specimen of pyrolusite from an unmentioned locality. 



The majority of the localities affording pyrolusite in this pro- 

 vince are almost certainly known to belong to the lower carboni- 

 ferous beds ; the country-rock of the ores has not in all cases been 

 made known, I saw last summer, in a locality about five miles 

 from the quartz and manganite conglomerate before mentioned, 

 which may be of New Red Sandstone age, a hard highly siliceous 

 rock, apparently quartzite (contiguous to slate), from which about 

 a ton of ore, consisting of pyrolusite and psilomelane, had been 

 recently taken. 



Wad. — This is found in various parts of the province, some- 

 times in abundance. One specimen, of black colour, from a con- 

 siderable bed situated, I believe, to the east of Halifax, gave me, 

 when dried at 212°, 56 per cent, binoxide of manganese, a great 

 deal of iron, a little cobalt, and a large quantity of insoluble 

 matter. In specimens of brown " paints " I have found from 11 

 to 20 per cent, binoxide of manganese, the greater part of the 

 residue being water and peroxide of iron. 



XXIII. On some Problems in Chances. By J. M. Wilson, M.A., 



Fellow of St. John's College, Cambridge, and Mathematical and 

 Natural Science Master of Rugby School*. 



THE problem of determining the probability that, if four 

 points be taken at random in an infinite plane, one of the 

 four shall lie inside the triangle formed by the other three, has 

 now acquired some degree of notoriety. Various solutions have 

 been given; \, \, § have all been obtained as results; and all 

 the methods are considered fallacious from introducing a com- 

 parison between infinities, and from those infinities apparently 

 having different relative values according to the mode of ap- 

 proaching them. 



I shall venture to offer a solution which depends on a different 

 principle, and I shall first illustrate it by applying it to a simpler 

 question. 



To determine the probability that if three lines are drawn at 

 random in an infinite plane, a fourth line drawn at random will 

 intersect the triangle formed by the other three. 



The peculiarity of this class of questions is that, if the triangle 

 is supposed drawn, it must be finite in conception as compared 

 with the infinities on all sides of it, and the chance required 

 would appear to be indefinitely small. But since the first three 



* Communicated by the Author. 



