150 



KEPORT — 1882. 



rection be required for a neighbouring point, N". Let A' B' be an approxi- 

 mate, and AB the true correction curve. (See figure below). 



Let A' M = ^1 (M) A M = ^aC^^). and so on. 

 Then, if A A' = a, B B' = ^, it is evident that 



^,(M) - f,(N) = f,(M) - ^,(N) + a 



/3. 



M 



N 



Now, in the method of approximation, the assumption made in the 

 first approximation is that the right-hand side of this equation is zero, or 

 that frXW) = ^.(N). 



A nearer approach to the truth is reached by neglecting a — /) only, 

 and obtaining the values of ^i(M) and '^i(N) from an approximate 



curve. Since the two curves in general 

 differ but little, and M and N are always 

 very near together, the error thus intro- 

 duced is obviously in general less than that 

 caused by neglecting ^i(M) — 0i(N). In 

 other words, since the curves are nearly 

 parallel, the diSerence between A A' and B B' 

 is in general much less than that between 

 M A' and N B', or M A and N B. 

 — Theoretically, of course, the error due to 

 the neglect of a — /3 would be greatest when 

 the two curves sloped in opposite directions in the neighbourhood of M N". 

 The only case, however, where a doubt as to the direction of slope could 

 arise would be near to a maximum or minimum, when the whole error 

 would be so small as to be negligible. 



This method, which is employed by Marek (he. cit.), is illustrated in 

 the case of Rudberg's and other methods in Part II. Since by it the 

 corrections at points are determined from measurements made on threads 

 whose ends lie near to but not actually at them, the corrections may be 

 regarded as transferred by means of the preliminary curve from the 

 observed to the theoretical points, and it is convenient to refer to it as 

 the method of transference. 



If the theoretical points are u and i, if the extremities of the thread 

 lie at M + Am and i -f- A?', and if <l>{p.i) represents the correction at m, and 

 so on, the true thread length is 



« + Am + ^ (m + Am) — [i + Ai + ^ (i + At)} . 



Here u + \u — (J, + A?!) = t say, is the uncorrected thread length, 

 and thus the true thread lensrth 



= I < + { ^. (m + Am) - ./. (m) } - { ^ (i + aO _ ^ (i) } I + d> (m) - </, (i). 



The corrections within the large brackets being determined from the 

 approximate curve, the expression contained in it is in this method used 

 instead of t, and is referred to as the transferred thread length. 



(7) In estimating the value of any method, three points require con- 

 sideration — viz. the accuracy attainable, the time and labour required, 

 and the cost of the necessary apparatus. The more accurate the appa- 

 ratus used, the simpler may be the method of calculation employed. 

 Thus, the chief objection to the step-by-stejj and subdivision methods is 

 that the errors are cumulative, and that by an unfortunate combination 

 of successive errors of the same sign, considerable inaccuracy may be 

 introduced. It is, however, evident that this objection would vanish if 



