430 PHILOSOPHICAL TRANSACTIONS. [aNNO I767. 



duct of the respective probabilities of those two events multiplied together ; if 

 therefore we multiply the two last-mentioned complements together, we shall 

 have the probability that some two stars would be, one within the distance x, and 

 the other within the distance z from the given star; and the complement of this 

 to unity, will be the probability that it would not be so : let us now suppose - to 

 represent this last quantity, and because the same event may as well happen in 

 respect to any one star as any other, multiplying this quantity into itself n times, 



according to the number of the stars, we shall have (j) representing the proba- 

 bility that no where, in the whole heavens, would be found any two stars, one 

 within the distance x, and the other within the distance z from the same star. 



If now we compute, according to the principles above laid down, what the 

 probability is, that no two stars, in the whole heavens, should have been within 

 so small a distance from each other, as the two stars jS Capricorni, to which we 

 shall suppose about 230 stars only to be equal in brightness, we shall find it to 

 be about 80 to J . For an example, where more than 2 stars are concerned, we 

 may take the 6 brightest of the Pleiades, and, supposing the whole number of 

 those stars, which are equal in splendor to the faintest of these, to be about 

 1500, we shall find the odds to be near 500000 to 1 , that no 6 stars, out of that 

 number, scattered at random, in the whole heavens, would be within so small a 

 distance from each other, as the Pleiades are.* 



* The computation of these probabilities are as follow : The distance between the two stars /3 Ca- 

 pricorni is something less than 3'-^; according to the rule above laid down therefore, if we suppose 

 230 stars equal to these in brightness^ the probability that no two stars of that brightness will be 

 found, any where in the whole heavens, within the distance of 3'^ from each other, will be repre- 



,^^r- ,6S75.5^ - 3.33 &c.» , 230 x 230 „ 

 sen ted by the fraction ( jr- . ) . irom twice the logarithm of 68/ a, 5 



[7.6746086] then, subtract twice the log. of 3.33 &c. [I.04.'i74i)6] and the remainder 6.0288590 

 will be the log. of the number of times, that 3.33 &c. ' is contained in 687.5.5', viz. -1254603 times, 



, 4254602 230 X 230 .„ , . , . . . . ^ , , 



and consequently ( c,-±f:n'i ' equivalent to the former fraction. From the log. 



of 4254<')d2, subtract the log. of 4254603, and the remainder will be— .0(0000102, the propor- 

 tional part of a unit in the number 4254603: this multiplied into 230 times 230, or 52900, gives 



— .0053958, the log. of the whole quantity, which corresponds to (he proportional part of a unit 

 between 80 and 81 ; this quantity is therefore equivalent to the fraction 



80 . . 1 



— nearly, the complement of which to unity is — . 



81 ' «1 



In the Pleiade.i, the five stars Taygeta, Electra, Merope, Alcyone, and Atlas are respectively at 



the distances of 1 1, 19j, 24j, 27, and 49 minutes from the star Maia nearly. From 7.67460 6, 

 twice the log. of ()»75.5, then, as before, subtract 2.0827S54, twice the log. of 11 (the number of 

 minutes between Taygeta and Maia) and the remainder 5.5918232 will be the log. of the number of 

 times, that 11^ is contained in 6875.5*, viz. 390682 limesj consequently a fraction, whose deno- 

 minator is this number, and whose numerator Is this number less by a unit, multiplied into itself 

 1500 times, will represent the probability, that no star out of 1500, scattered by chance in the 



