578 LAPLACE. 



1010. Laplace now proceeds in his pages 822 — 329 to the 

 case where two unknown elements are to be determined from a 

 large number of observations ; see Art. 923. Laplace arrives at 

 the conclusion that the method of least squares is advantageous 

 because the results which it gives coincide with those obtained by 

 making the mean values of the positive errors to be apprehended 

 as small as possible ; the investigation is very laborious. The 

 same assumptions are made as we have stated at the end of 

 Art. 1007. 



Laplace considers that he has thus established the method of 

 least squares for any number of unknown quantities, for he asserts, 

 on his page 327, ... i7 est visible que V analyse precedente 2')eut seten- 

 dre a un nomhre quelconque d'elemens. This assertion, however, 

 seems very far from being obvious. ' 



Poisson has not considered this part of the subject ; on account 

 of its importance I shall now supply investigations by which the 

 conclusions obtained in Art. 1007 will be extended to the case of 

 more than one unknown element. I shall give, as in Art. 1007, 

 two modes of arriving at the result ; Laplace himself omits the 

 first, and he presents the second in a form extremely different from 

 that which will be here adopted. In drawing up the next Article 

 I have obtained great assistance from the memoir by R. L. Ellis 

 cited in Art. 1001. 



1011. Suppose that instead of one element to be determined 

 by the aid of observations we have any number of elements ; sup- 

 pose that approximate values of these elements are known, and 

 that we have to find the small correction which each element 

 requires. Denote these corrections by x, 2/, z, ... Then the general 

 type of the equations furnished by the aid of observations will be 



€i = aiX + hiy^CiZ+ ... -qi ( 1 ) . 



Here 6^ is unknown, while a^, hi, c^, ... q^ are known. Multiply 

 (1) by 7i, and then form the sum of the products for all values of 

 ^, which we suppose to be from 1 to s, both inclusive. And let the 

 factors 7^, ^y^, ... 7^ be taken subject to the conditions 



27A=0, :^7A = 0, (2); 



