244 Mr. Ivory on the Methods proper to be used for deducing 

 then, by adding all the equations, and observing that, 



._ e sin* X -\- e' sin - x' + e" sin - x" + &c. 



Bcr + Q = ' 



= n 



we shall get, 



B. A + C.f- D = Bcr + g. 

 Having now two equations containing A and f, we obtain 



from them, 



AC-BD , B ? 



C-B L li fi 



/• - D ~ AB . g 



./ C-B- " 1 " C-B 2 * 



It is manifest from the foregoing analysis that the suppositions 

 o" = 0, g = 0, correspond to the two equations, 



(Z e , d e' i, (I e" n 



;l A </A a A 



d e , de' ,, dc" n „ 



which determine the absolute minimum of the sum of the 

 squares of the errors with respect to the variables A andy." 

 Hence if we make § = 0, cr == 0, in the formulas (B), the re- 

 sulting values of A and f will coincide with those found by 

 the method of the least squares. In general it appears that 

 we cannot avoid an error in A equal to <r, that is, to the 

 mean of all the errors of the pendulums. But it is chiefly the 

 terms containing g that produce great changes in the relation 

 of A and f, and alter notably the figure of the earth. The 

 reason is that the divisor C — B 2 is always less than unit, and 

 in many cases only a small part of unit; hence the terms in 

 question are considerable, even when § itself is very small ; 

 and as they have opposite signs in A and f, they have great 

 effect in altering the ratio of those quantities, and consequently 

 in changing the ellipticity. This argument is quite conclu- 

 sive against the use of the method of the least squares for the 

 purpose of deducing a mean result from a number of pendu- 

 lum experiments. 



In order to illustrate what has just been said, we may take 

 the example before alluded to, at p. 334- of Captain Sabine's 

 work: then A = -09107 



B = -372977 



C = -292646 



D = -065006 ; 

 which numbers being substituted in the formula? (B), we shall 

 get, A = -01568 + cr — 2*429 x? 



/= -20213 -j- 6-513 X f . 

 The suppositions t = and § — give the solution of Cap- 

 tain 



