P 11 OB ABILITY 781 ment, personal error of the observer, and others. The error of the observation is in fact the sum of the partial errors arising from these different sources ; now these evidently cannot be taken each to follow the same law, so that we have here a more general problem of the same species, viz., to combine a number of partial errors, each having its own law of facility and limits. There is every reason to suppose that the error incurred in any single observation or measurement of any kind is generally due to the operation of a large number of independent sources of error ; if we adopt this hypothesis, we have the same problem to solve in order to arrive at the law of facility of any single error. 48. We will consider the question as put by Poisson (Rechcrches, p. 254 ; see Todhunter, History, p. 561), and will adopt a method which greatly shortens the way to the result. Let x be the error arising from the combination or superposition of a large number of errors e 1( .,, e 3 . . . . each of which by itself is supposed very small, then x = e 1 + e 2 + 3 + (37). Each partial error is capable of a number, large or small, of values, all small in themselves ; and this number may be quite different for each error e l f. 2 , f 3 There may be more positive than negative, or less, for each. 1 If n lt n.,, n s .... be the numbers of values of the several errors, the number of different values of the compound error x will be Wjn^ij . . . We will suppose it, however, to take an indefinite number of values N, some multiple of the above, so that the n l} n. 2 , n. A .... different values are repeated, but all equally often, so as to leave the relative facility of the different values unaltered. We will suppose the same number N of values in every case, whether more or fewer of the partial errors e lt e.,, e 3 . . . . are included or not. Let the frequency of an error of magnitude x be called y, and let the equation expressing the frequency be y=f(x) (38); i.e., ydx = number of values of x between x and x + dx. The whole number of values is where /n, /j. are the sums of the higher-and lower limits of all the partial errors. If now a new partial error e be included with the others, let it have n particular values c, e, e" ; if it had but the one value c, then to every value x of the old compound error would correspond one x of the new, such that x + e x ; and the number of values of the new from x to x + dx is the same as of the old from x to x + dx that is, f(x)dx, or f(x - e]dx . Now the next value c gives, besides these, the number f(x c )dx , and soon. Thus the whole number of values of the new compound error between x and x + dx is / fix c) + f(x c ) + fix 1 c") + . , dx . Hence the equation of frequency for the new error is (dropping the accent, and dividing by n that is, reducing the total number of values from X?i to N, the same as before) y = n-i{f(x-e)+f(x-e )+J(x-c")+ . . .} . . (39). Hence y-f(*) - e+e>+e + -f(x) + f +e " +e " 2 ^^f"(x) , It Ib neglecting higher powers of c, e . . . . Hence if a new partial error , whose mean ralue = a, and whose mean square is A, be superposed onthe compound error (38) resulting from the combination of a large number of partial errors, the equa tion of frcquancy for the resulting error is It thus appears that each of the small errors only enters the result by its mean value a, and mean square A. If a second error were superposed, we should thus have y = (1 - BI D + iAjD -Xl - aD + UD 2 )/(.T) ; as A is a lower infinitesimal than a, we retain no other terms. - (a+ qJD + A + A/ - a i 2 + (a + ^ )2 D-; /(*). Thus any two errors enter the result in terms of a + a 1 and A + A - a 2 - c^- ; as this holds for any two, it is easy to see that all the partial errors in (37) enter the equation of frequency (38) only in terms of m and h-i ; putting ?H = a 1 + a_, + a 3 + . . . = sum of mean errors, /(. = l + ., + A 3 + . . . =sum of mean squares of errors, J- (41). i = a + a: + al + . . . = sum of squares of mean errors, 1 An error may have all its values positive, or all negative. In estimating the Instant when a star crosses the meridian we may err in excess or defect, but in estimating that when it emerges from behind the moon, we can only err in excess. We have heard this instance given by Clerk Maxwell. Thus, y=f(x)-=$(x, m,h- i) . Let m receive an increment 8m ; this is equivalent to superposing a new error whose mean value is 8m, and mean square infinitely smaller (e.r/., let its values be all +, or indeed we nJay take it to have but the single value 5m) ; ... 8w-^5m- - ^-Sm, by (40); dm dx il ^L = _ dy dm dx Hence y is a function of x-m ; so our equation must be of the form y = Y(x-m, h-i) (42). Let h receive an increment Sh ; or conceive a new error whose mean value a = 0, and whose mean square = 5/t ; we have (40) Hence (43). dx- dh Let us now suppose in (37) that all the values of every error are increased in the ratio r ; all the values of x are increased in the same ratio ; consequently there are the same number of values of x from rx to r(x + dx) as there were before from x to x + dx. This gives Y(x m,h- i)dx = J?(rx rm , r-(h - i))rdx , for m is increased in the ratio r, and h and i in the ratio ? i2 . Let us write for shortness | = x - m , r] = h i, so that 2/=F(|,7)) (44); we have ? ~ 1 F(|, 77)=F(r|, r-ri) . Let r = l + o>, where o> is infinitesimal ; This equation, and i = 2-- " 7J identical with (43), contain the solution of the problem. Thus, (45) gives by integration (45). (46), (47). Again, combining (45) and (46), =0; Substitute for y the value (47); and we find that is, a function of |TJ - * identical with a function of 77. This cannot be, unless both sides are constant. Hence TJ-| + ?/ = c. Xow c = 0, for ^ vanishes with |, by (47); and, y being always finite, the left hand number vanishes with ; T Substituting in (47) and restoring the values of {, 77, we find the form of the function (42) to be ij^C(h-i)-^e-2(i^i) (48)- C is a constant depending on the number X. The probability of the error x falling between x and x + dx is found by dividing ydx by the whole area of the curve (48) ; i.e., p (49). 49. If, instead of eq. (37), we had put a- = 7] e i + 72 f 2 + 73^+ ....... ( 5 ) where y lt 7 2 , 7., .... are any numerical factors, the formula (49) gives the probability for x, provided h, i, m are taken to be instead of the values in (41). 50. If we take the integral of eq. (49), between any two limits
H, v, it gives us the probability that the sum x of the errors lies