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



signs, according to their absolute magnitude, and begin to 

 reckon from the smallest, then the error which belongs to the 

 index ^ (m + 1), if m is an odd number, — or the arithmetical 

 mean between the errors of which the indices are ^ m, and i m 

 4- 1, if wi is an even number, — will give an approximate value for 

 r. In the example given above, m being = 29, it would be the 

 15*^, or we should find 



r = 5"'-914. 

 As in the sums of the powers a greater number of errors so 

 greatly increases exactness in respect to the probable limits, the 

 effect must be so much the greater in this mode of computing. 

 As the necessary formulahas been given without proof, by Gauss, 

 in the Zeitschrift fur Astronomie, vol. i. p. 195, the following 

 elegant demonstration, for which I am indebted to my respected 

 colleague. Professor Dirichlet, will have the more value, as 

 the proposition has not yet been demonstrated elsewhere. 



Let us seek the probability that, with (2 ti + 1) observations, 

 the distribution of the errors shall be such, that there shall be 

 one error between t and t + d t, n errors between and t, and n 

 errors between t + dt and co . Let the probability that there is 

 one error less than t be generally 

 t 

 = r^{A) d A =u; 



then the probability of an error > t + d t will be 



\—^tdt- r\> A d A = l-u-^tdt, 



as the probability of an error between t and i + d t = ^ {t) dt. 

 Hence the compound probability of an arrangement of errors 

 in which n errors < t, one error between t and t + dt, and n 

 errors > t + dt. 



= tt" (l-^<)".^t/ {t) dt, 

 neglecting the members of the second order, as the result is of 

 the first order. But there may be as many such cases or arrange- 

 ments as there are possible transpositions of 2 ?^ + 1 elements, 

 if there occur among them n equal elements of one kind (of 

 which the probabihty = u), and n equal elements of another 

 kind (of which the probability = (1 —?/)). Consequently the pro- 

 bability of all possible aiTangements of this kind 



