SOME MERCUHV sTANl>Ai;l>s OF KKSISTANCE, ETC. 59 



In general, however, the thread length \s not always entirely situated within that 

 portion of the tube afterwards chosen for a resistance standard. Let a.b.c . . . &c., 

 represent the fractional parts of the length situated within the region mentioned, and 

 L be the length of the resistance standard. Then, if we assume, in the first instance, 

 that the successive portions of the tube occupied by the thread are each uniform in 

 cross-section, the mean section s of the length L will be such that 



= (2W/AX + rtW/AA,, + 6W/AX* + . . . &c.)/(n + + & + ...), 



where n represents the number of complete lengths, and X, X, A,, &c., those lengths 

 but partly situated within the limits. 



The resistance R of a column of mercury at C. of length L and of uniform 

 section s is equal to pL/x, where p is the specific resistance of mercury. 



Similarly the resistance R' of (n + n + & + ) tubes, each one of constant cross- 

 section, and having a total length L, is given by the equation 



whence 



R' 

 R 



p, _ /- L/n + . . . , L/n + . . . i &L/n + . . . 

 Bf r W/AX"" "W/AX. ~W/AX 4 



(SX + aX ( , + feX 4 + . . .) (Sl/X + a/X~ a + 6/X, + . . .] 

 ( + a+:*+...)F 



This ratio is the first approximation to the correction for conicality. In the sequel 

 R'/R is denoted by //. The calibration was such as to enable five different series to 

 be chosen, the resistance of each series being that of the standard. 



There is no justification, however, for assuming that each length L/n is uniform in 

 section, a*nd a correction has still to be applied for their conicality. The nature of 

 this latter is, however, known to a sufficient degree of accuracy from the calibration 

 curve. For, in the case of a tul>e of slightly conical bore and of uniformly varying section, 

 the mean section for a portion, as determined by the length and mass of a mercury 

 thread, may be regarded as the actual section at the mid-point of the thread. Thus, 

 with a conical thread 5 centims. long, and such that the terminal sections differ by 

 2 per cent , the mid-point of the thread is but 0'02 mi Him. distant from that section 

 equal to the mean. The irregularities of the 5-centim. lengths are, therefore, 

 approximately portrayed by the calibration curve. The discrepancy is greatest for 

 those positions where the bore is not uniformly conical, and an idea of uniformity is 

 conveyed which is greater than really exists. Such irregularities, however, are 

 insignificant for the present purpose. 



The resistance of a conical rod of conducting material being calculable, it becomes 

 easy to evaluate the conical corrections for the 5-centim. lengths, since these latter 

 are of conical structure. The calculation is as follows : 



If dx be a small element of length, and s the corresponding cross-section, then the 

 resistance R of a slightly conical mercury column may be taken as p\dx/if. For a 



I 2 



