J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 323 



For the first proposition the calculus of probabiUties gives the 

 following expression. 



I. 



If (f> ^ is the probability of the error A, <f) A' that of A', and 



so on, then the probability of the concurrence of the errors 



A, A', A", &-C. is 



= </> A . A' . (^ A" . . . 



We may convince ourselves of the truth of this in the fol- 

 lowing manner. Let us assume, for instance, that in three 

 observations the error A be found twice, and the error A' 



once ; further, let ^ A =-^, <f) A' =-^. Let the three observa- 

 tions be regarded as belonging to a series of observations, m, so 

 extensive that in it all errors shall occur according to their pro- 



babiUty ; consequently, — m errors equal to A, — w errors equal 



to A', will occur in it. Let the number of the remainder be s, 

 in which it is here indifferent how many equal or unequal there 

 are among them. Apart from s, the number of all possible ar- 

 rangements of the errors in the m observations will be 

 1 . 2.3 ...OT 



1 .2 .3 ...^m. 1.2.3... ^ m. 

 n n 



As three places are taken up by the two A and one A', there are 

 left for the remaining m — 3 observations, 



1.2.3 (m - 3) 



1.2.3 



/£^_o Yl .2.3.0.. /^^m-1 ) 



I. Consequently the pro 

 two A and one A' should 



\n J n n 



possible mutations. Consequently the probability that in any 

 three observations two A and one A' should be found 



-^ m 



(m — 2) . (m — 1) .m 

 or 



(P-- — \ . ^ . -^ 



_ V?? m ) ' n ' n 



-(,_n.(,_i)., 



The assumption on which we have proceeded is, however, 

 strictly true only for m = oo , or the probability of a single com- 



