VOt-'LVII.] PHILOSOPHICAL TRANSACTION?. 429 



position that they had been scattered by mere chance, as it might happen. Now 

 it is manifest, on this supposition, that every star being as likely to be in any 

 one situation as another, the probability that any one particular star should hap- 

 pen to be within a certain distance, as for example one degree, of any other 

 given star, would be represented, according to the common way of computing 

 chances, by a fraction, whose numerator would be to its denominator, as a circle 

 of one degree radius, to a circle whose radius is the diameter of a great circle, 

 this last quantity being equal to the whole surface of the sphere, that is, by the 



fxtV 



fraction ( ^„_, ,, )% or, reducing it to a decimal form, .000076154, that is, about 

 ^ 0875.5 ' '^ 



J in 13131; and the complement of this to unity, viz. . 999988846, or the 

 fraction -, will represent the probability that it would not be so. But be- 

 cause there is -the same chance for any one star to be within the distance of one 

 degree from any given star, as for every other, multiplying this fraction into itself 

 as many times as shall be equivalent to the whole number of stars, of not less 

 brightness than those in question, and putting n for this number, .999928846", 



13130 



or the fraction (TT-.-rr)', will represent the probability, that no one of the whole 

 number of stars n would be within one degree from the proposed given star; and 

 the complement of this quantity to unity will represent the probability, that there 

 would be some one star or more, out of the whole number n, within the distance 

 of one degree from the given star. And further, because the same event is 

 equally likely to happen to any CMje star as to any other, and therefore any one 

 of the whole number of stars n might as well have been taken for the given star 

 as any other, we must again repeat the last found chance 11 times, and conse- 

 quently the number .999923846''", or the fraction {tt\-t7-T will represent the 



\.ij Xo Xil 



probability, that no where, in the whole heavens, any two stars, among those in 

 question, would be within the distance of one degree from each other ; and the 

 complement of this quantity to unity will represent the probability of the con- 

 trary. 



By a like reasoning, if we would compute the probability, on the same suppo- 

 sition, that no two stars should be, one within the given distance x, and the 

 other within the given distance z, of some one particular star, we must first find 

 the probability that no star, of the whole number of stars n, would be within the 

 distance x from the given star, which will be represented, as before, by the frac- 



6875 5*— XX 

 tion (— > -■ > ,— )"; and secondly, we must find the probability that no star, of 



the whole number of stars w, would be within the distance z from the given star, 

 which, for the same reason, will be represented by the fraction ( -~ — ~^)"; and 

 the complements of these to unity will represent the respective probabilities of 

 the contrary; but the probability that two events shall both happen, is the pro- 



