ERRORS OF OBSERVATION. 423 



Further, it appears that the average of an unlimited number of observations, under one 

 law of facility, must be the most probable value; must be in fact the true value. For if t 

 be the true value, the average of a observations is t + (7"'2e; and ct~' and 2e both vanish. 

 This can also be proved from the development of 2\|/'(a — a?) = 0. And next, we know that 

 any observations which, independently of their value and number, give the average, weighted 

 or not, as the most probable value, must be made under the modulus ^c, 6"'^'^':^/^. 



Now let there be any number of sets we please, each of an infinite number of observa- 

 tions, ffi, 0-2, &c. in number. Let each of the averages, Mi, M^, &c. be held an ob- 

 servation. We know that the average of the whole is the most probable result, namely, 

 S(crJlf) : 2(7, independently of the number of sets of observations; consequently, the modu- 

 lus of each is of the form asserted; but each is the average of an infinite number of 

 observations. This argument is subject to the difficulty that J/,, M^, 8ec. are equal, being 

 each of them the truth in question. But if, instead of supposing the observations infinite 

 in number, they were to be taken as only very great, and the several parts of the reasoning 

 asserted approximately, instead of absolutely, the whole would become a demonstration of 

 that kind which, though far from satisfactory, is cogent enough to throw doubt upon any 

 contradictory conclusion, however arrived at, until absolute fallacy is detected. And this 

 will never be done; for all the steps are substantially true, though requiring the introduction 

 of limits for their explanation. 



I shall not enter upon the special points connected with the method of least squares, in 

 the common case in which the functions P,, Pj, &c. are of the form aai + by +... To this 

 form all cases will be reduced in practice : for when we deal with d)(a?, y, ss) we generally 

 know approximate values of <r, y, is. If <r =a;i + f, &c., where ^, &c. are small, our function 

 takes the form A + B ^ + &c., powers and products being rejected as inconsiderable. And ^, 

 8ec. become the subjects of discussion. 



The probable — or critical — error depends more upon the universal use of the final 

 modulus, -y/c e'""' : ^/ir, than any other part of the subject. No attempt has been made 

 to halve any curve of error except the final curve. The modulus being (px, an even function, 



and 2 / (pwdx = 1, the probable error is, approximately, 



40 6.4^0* 12.4»0'V^" 10/ U.4,'(p">V" 5 280/' 



where 0, 0^^, 0,^, &c. are the values of (p,v, (p"a>, (fi"'x, &c. when x = 0. The all important 

 theorem that the square of the probable error of the sum is the sum of the squares of the 

 probable errors of the aggregants, is entirely the property of the final modulus. We see 

 that it is not lost in the second approximation : but it would not remain true in the third. 

 The mode of assigning the quantity of the probable error is the most unsafe part of the 

 first approximation; that is, of the simple use of the final modulus. It may be wrong, as 

 the second approximation shows, in any proportion between that of 8 to 8 and that of 8 to 11. 

 When any use of caution is made of the magnitude of the probable error, it would be 

 Vol. X. Part II. 54 



