Mr. ELLIS, ON THE METHOD OF LEAST SQUARES. 215 



neglect for instance every such term as -— :j ii^'', because we have a term in ii''' divided by «''. 



71-'' u-'^-^^ 



Thus we retain -^ and neglect — — -, although, unless ii be large, the former term is of the samt 



or a lower order of magnitude than the latter. That Laplace's method does in a very general 

 manner give an approximation of this kind cannot, I think, be questioned, especially after the 

 verification we have just gone through. But some doubt may perhaps remain, whether such an 

 approximation to the form of the function P, if such an expression may be used, is also an 

 approximation to its numerical value, when we consider that in obtaining it we have neglected 

 terms demonstrably larger than those retained. 



For two recognized exceptions to the generality of Laplace's investigation, viz. where 



<pe = - 2' ^"'1 ^^^ '^^^^ '" which //,, H2---, decrease in infinitum sine Umite, I shall only refer 



to p. 10 of Poisson's paper in the Connaissance des Terns for 1827. Neither affects the general 

 argument. We now come to Gauss's second method, which is given in the Theoria Combinationis 

 Observationum. 



GAUSSS SECOND DEMONSTRATION. 



The connexion between the method of Laplace, and that which Gauss followed in the Theoria 

 Comhinationis Observationum, will be readily understood from the following remarks. 



After determining /u,...;u„ by the condition that P should be a minimum, Laplace remarked 

 that the same result would have been obtained (viz., that 'E/u.^k'' must be a minimum), if the 

 assumed condition had been that the mean error of the result, i. e. the mean arithmetical value 

 of 2/U6 should be a minimum. (I should rather say that he makes a remark equivalent to this, 

 and differing from it only in consequence of a difference of notation, &c.) It is in fact easy to see 

 that the mean value in question is equal to 



•'" , , or to 2_/(, 7ipdu; 



Jo Pdu 



and as 



P = 



2 (^Sm'A'^)' 



^« , 2 (2^=A-=)' 



2 / updu = y= 



•^» V-n- 



which is of course a minimum when 'E/j.^k^ is so. 



Gauss, adopting this way of considering the subject, pointed out tiiat it involved the 

 postulate that the importance of the error 2/if, i. e. the detriment of which it is the cause, is 

 proportional to its arithmetical magnitude. Now, as he observes, the importance of the error 

 may be just as well su])poscd to vary as the square of its magnitude : in fact, it does not, strictly 

 speaking, admit of arithmetical evaluation at all. We must assume that it is represented by some 

 direct function of its magnitude, such that both vanish together. One assumption is not more 

 arbitrary than another. Let us suppose, therefore, that the importance of the error is repre- 

 sented by (2;ic)". That is, that (2/uf)'' is the function whose mean value is to be made a 

 minimum. I now proceed to find it. 



C^mO' = ^I'-'C + '2-2,x,^,e,e, (13). 



