SOMK MKI.Vl i;\ STANDARDS OF RESISTANCE, ETC. 63 



obtained by the use of a planimeter or by calculation of tbe mean X. Let L, denote 

 the length of the standard portion AB as measured along the axis, and S, be its mean 

 cross-section. Let the limits of the mercury column employed to obtain a value of 

 L//W.J be denoted by C and D. The positions of C and D will vary slightly with 

 the different measurements, but will always closely approximate to the limits A and B 

 respectively. Also let L 2 denote the length CD as measured along the axis, and S 2 

 be the mean cross-section of this length. 



Then the length L, is obtained by the algebraic addition of the two short lengths 

 AC and BD to Lo. Let these short lengths be denoted by /' and /". The mean 

 cross-sections of these in terms of S, may be obtained from the positions of their 

 mid-points and b relatively to the zero line. Let them be written a-S, and- yS, 

 respectively, where x and y are dependent on the ordinates of a and b. In 

 consequence 



S, = (L,S 2 + /'a-S, + ryB,)/(L, + V + /") ; 

 that is 



s,/s 3 = IVJL, + v (i - x) + 1" (i - y);-, 



so that 



From this expression it is seen that the length and position of L, is Ixjst chosen so 

 that V shall be nearly equal to I" ; also, if x and y lie of opposite signs, /' and I" should 

 be of the same sign, and rice versA. 



In general, L, is so nearly equal to L, that the expression I' (I x) } I" (1 + y) 

 is very small, and approximate values only of x and y are required. 



Condition of Axis of Tube. 



A matter of some importance is the curvature of the axis of the tube in different 

 parts of the standard. The definition of the international ohm assumes the axis to 

 be perfectly straight, but in practice it can only be approximately so. The cross- 

 section varies considerably, and it is only reasonable to suppose that the axis (i.e., 

 the line joining the central points of the cross-sections) should also vary from the 

 straight line. That the curvature in any one portion of the tube is not great is 

 obvious by inspection, but there is no certainty from inspection that it may not 

 considerably affect the resistance of a mercury column. 



As an example, consider an axis of undulating form, and, for simplicity, let it be 

 divided up into a number of arcs of equal curvature. Then for one of these, say 

 AB, the length measured for the determination of L 9 /W will be the straight line AB, 

 in consccjucmv <>t' wliicli L is too small, and therefore the resistance of the mercury 

 column greater than that calculated. The true resistance of the portion AB may, 

 however, be evaluated in the following manner. 



