30 ON THE METHOD OF LEAST SQUARES. 



tion of this kind cannot, 1 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 <e=-- - - z , and the case in which 



/A 4 , //. 2 ..., decrease in infinitum sine limite, 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 Combinalionis 

 Olservationum. 



GAUSS'S SECOND DEMONSTRATION. 



The connexion between the method of Laplace, and that 

 which Gauss followed in the Theoria Combinationis Observa- 

 tionum, will be readily understood from the following remarks. 



After determining ^...^ by the condition that P should be 

 a minimum, Laplace remarked that the same result would have 

 been obtained (viz. that 2/& 2 * must be a minimum), if the 

 assumed condition had been that the mean error of the result, 

 ^*. e. the mean arithmetical value of 2/ie 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 nota- 

 tion, &c.) It is in fact easy to see that the mean value in ques- 

 tion is equal to 



and as 



. r , 

 2 Jo updu 



F* 



2 



,- , 

 VTT 



which is of course a minimum when 2//, 2 & 2 is so. 



