J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 363 



because, by virtue of the form for two unknown magnitudes, 

 the form for any number is at once derived, — if there are 

 three, by first combining two with each other, and then com- 

 bining their result with the third,— if there are four, by fii'st 

 combining three with each other, and then their result with the 

 fourth, and so on. 



The general problem might be solved in a similar manner if 

 the integrations could be performed. For 



X=/(^,^,^'....) (21.) 



the probability of the concurrence of arbitrary values of the (u. 

 variables will be 



TT 



If we are here to consider only the cases in which a determi- 

 nate value for X is to be found, let us express one of the vari- 

 ables .r as a function of X and the remainder. If we substi- 

 tute this value in the exponents, and take the sums or in- 

 tegrals within all possible limits for x, od . . ., we shall obtain 

 the probability of the value X, and we may thence determine 

 the most probable value and its limits. But for this the know- 

 ledge of the function is obviously needed ; and, if this function 

 is not linear, the complete integration will be impracticable in 

 most cases. However, under the supposition that the limits 

 for the several variables are already so narrow, that the higher 

 powers of the probable error may be neglected, we may find an 

 approximative value for X and its limits, which will be always 

 sufficient in practice. 



Let us take for arbitrary values of x, a^, x" . . . the form 

 a + A X, a' + A x', a" + A x" ', then, if 



Y=f{a,a',a"...) (22.) 



the general expression for X, neglecting those powers of A x, 

 A US', A a?", which exceed the first, will be 



X = V + 



Q--Q-^-(IJ)--'. 



and the probability of the concurrence of these values will be 



2 B 2 



