J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 361 



as a common standard, then the weight of a, by reason of its 

 probable error r, will be 



and likewise for a' 



.-J _ 

 r 



P - -ZF2' 



Hence, if h belongs to w, the probability of any value for x 



— ^ '^P „-mp(x-ar- 

 and for a?' 



_ ^ l/y g - h^ p' {x'- a'Y , 



thus the probability of the concurrence of two arbitrary values 

 will be 



and the probability of the concurrence of two values x and x' 

 which satisfy the equation 



X + x' = X, 

 in which X signifies an arbitrary but determinate value, is 

 found by considering one of the magnitudes x or x' as a func- 

 tion of the other, and of the magnitude X, and by substituting 

 the value so obtained. Hence the probability that any value x, 

 by its concurrence with the value x', should give the residt X, 

 is 



^ ^ h^ Vpp' ^_,fi {p(^_„)8+p/(x_x-a')'}. 

 w 

 Then, if we take the sums of all possible values of W, or the 



J Vf dx within the limits in which a value of x can exist, being 



here — x and + co , we shall have embraced all cases in which 

 X can be obtained, or have determined the probability of X. In 

 order to facilitate the integration, let us give to the exponents 

 the following form : 



which is at once obtained if we combine in a quadratic form all 



VOL. II. PART VII. 2 B 



