ERRORS OF OBSERVATION. 419 



of his Doctrine of Chances. Two extracts will show how nearly his forms approached to those 

 of Laplace. 



(p. 236.) ' I also found that the Logarithm of the Ratio which the middle Term of a 

 high Power has to any Term distant from it by an interval denoted by I, would be denoted by 

 a very near approximation, (supposing m = i «) by the Quantities 



m + I 



m + I - ^ ii log . m + I - 1 + m - I + ^ X \og . m - I + 1 - 2m y. log .m + \og ,' 



m 



(p. 242.) ' If, in an infinite Power, any Term be distant from the Greatest by the Interval 

 /, then the Hyperbolic Logarithm of the Ratio which that Term bears to the Greatest will be 



expressed by the Fraction — x U ; provided the Ratio of ^ to w be not a finite Ratio, 



but such a one as may be conceived between any given number p and \/n, so that I be 

 expressible by p ^/n, in which case the two terms L and R [equidistant from the greatest] 

 will be equal.' 



Let the modulus of facility be assumed to be 



= V-.e- 



(px= \/ — .e'"^ (p + qa^ + rx*+ sx^ ). 



Let it be found that a few at least of the observed averages A2, Ai, Sec diminish rapidly. 

 Let (2c)"' = h : then from / (px . x^dx = A^y, we find (^A^ being unity) 



i> + j/t + S.rh^Jr S.5.sh?+ = 1, 



p+ 3qh+ S.5.r¥+ 3.5.7.sA'+ = A.fi-\ 



3p + 3.5qh+ S.5.7.rh^+ 3.5.7.9.sA'+ = A.h'^, 



3.5p + 3.5.7 .qh+ S.5.7 .g.rh? + 3.5.7.9.Ush^+ = A^h'^ : 



and so on. If q, r, &c. be small compared with p, we may make successive approximations, 

 of which two will be sufficient, seeing that practice is well satisfied with the results of one. 

 But practice does not know that one reason of her contentment with the first approximation is 

 the practical accordance of the second with the first, in everything but value of constants. 

 This circumstance tends to lessen our surprise at that pliability of the function e''"' which has 

 been illustrated. 



For first approximation we have p = \, p - AJi~^, or (2c)"' = .^j. And we have 

 ^/c . €'"'' : -v/tt for the modulus. This is the well-known case : c is the weight of the 

 observation ; and the probable error — critical error would be a better name — is '476936 : y/c. 



; . The second approximation is obtained from 



p + qh = 1, p + Sqh = AJi''^, 3p + I5qh = AJi'^, 



3h - A, Ao- h 



or 3h^ - GAJi -^ A^= 0, p = 



2A ' ^ 2A^ 



