LAPLACE. 579 



thus we obtain 



«,=|m+|M 3^_ 



Now we know from equation (4) of Art. 1002 that there is the 

 probability 



irl}"^' W. 



that Syi^i will lie between Z — 2t/c and Z + ^tk, where, as before, 

 I = Syj^v Put Z for ^7^6^ ; thus (3) becomes 



^=|Mt+ ^ ..(5); 



and there is the probability (-i) that the error in the value of x, 

 when determined by (5), will Lie between 



2T/g 



We propose then to make ^ as small as possible, the fac- 

 tors being taken consistent with the limitations (2). 



Since it is obvious that we want not the absolute values of 

 the factors 7^, 73, 73, •••, but only the ratio which they bear to 

 any arbitrary magnitude, we shall not really change the problem 

 if we impose the condition '^'yia^ = 1. Thus, since k^ = ^j^h^, we 

 require that '^jiV shall be a minimum consistent with the con- 

 ditions 



Xyiai = l, tyihi=0, ^7^ = (6). 



Hence, by the Differential Calculus, we have 



tyjitdyi=0, 

 ^Gidyi = 0, 

 ^hid^i = 0, 



Therefore by the use of arbitrary multiphers \ fi, v, ... we 

 obtain a set of s equations of which the type is 



ry.Jli^ = \a. -\- fJ^hi + VCi + ,....(/). 



37—2 



