﻿192 ON SOME APPLICATIONS OF THE METHOD 



(5) and (8) are two of the elementary formulae in the method of mechanical quadra- 

 tures. Usually the convergence of the differences of Fand F, and the smallness of 

 their coefficients, allow of the differences above the second or third order being neglected. 



It is an objection which applies with some force to the practical use of this method, 

 that a numerical error committed in any part of the work is, by the nature of the pro- 

 cess, disseminated over a great mass of computations. The best safeguard for accura- 

 cy is the use of suitable tests and checks at proper intervals. The chief expenditure of 

 labor will generally be in computing the series of numerical values of V and F; any con- 

 siderable accidental errors in these quantities will show themselves in taking the dif- 

 ferences. As these quantities are independently calculated, there is no tendency here 

 to an aggregation of errors. But in the summation of V or F, an error of addition in 

 forming any one value of A affects also those succeeding it. A security against this 

 will be to use (6) and (9) after A it has been independently found by the successive 

 addition of A\ A' n _ t . 



It is useful to know what degree of accuracy in For F will be sufficient, in a given 

 case, to compute A n accurately to a certain number of decimal places. If J be known 

 to the same number of decimal places (not significant figures) with A, and in every in- 

 stance has an error e in its last decimal place, and always with the same sign, then 

 the possible error of A u =ne. And a similar error e' in F gives the possible error 

 of A. = "-^p e'. 



But it is plain that these possible errors are highly improbable. Regarding it as 

 equally probable that the last figure of each value of V or F is too large or too small by 

 one unit, or exactly right, when the number of intervals or n is given, the comparative 

 probabilities of the errors of A ri , or of A\ — J y , being within certain limits, will be near- 

 ly expressed by 



I 2 . 2 2 . 3 2 V 



1.2.3 a. 1.2.3 b' 



where a and b are respectively the number of times that the last decimal place of V or 

 of F must be too large and too small by unity, in order to produce the given errors in 

 A n or J\ — z/ ' ; the whole number of instances in which V or F will be too large or 

 too small being assumed as about two thirds of n, or a + 6 = In, and /t = T«= a 4" i ) 

 a, 6, and h being the nearest whole numbers to their exact values, and unity the prob- 

 ability that a = b, or that the error of A n = 0, which is its most probable value. 



\Vhen A n — A„, or A\ — z/J, increases pretty uniformly, f^will be of the same order 

 with 4- (A a — A^) and F of the same order with 4- (A l n — A l ), and consequently the 

 number of significant figures in Tor F will usually be much less than in A. But these 



