G4 



MR. F. E. SMITH ON THE CONSTRUCTION OF 



Fig. 4. 



If we aasume the curved portion to be of uniform cross-section, then the curvature 

 will be different at different parts of the surface, and the surface may be imagined 

 to be described by the motion of radii of different lengths, all parallel, and all 

 emanating from a straight line. In fig. 4 the radii vary in length from 7^ to r 2 , the 

 mean radius of curvature being a. Let the angular motion of the radii be a. Then, 

 if r be any one of the radii, the length of the corresponding element of the tube is 

 rot, and the cross section of the element (see the figure) is dr\/(r rj (r 2 ?). 

 Hence, if p be the specific resistance of mercury, and II the resistance of the column 

 under consideration, 



1 If//- -.dr 



R-- | v X ('' -')('**- 





If b be the radius of the cross-section of the tube, and x = h sin 0, 

 It" 1 = (pa)" 1 !^ 2 cos 2 6ddl(a + b sin 0). Inserting the limits and 2ir, and 



/ I ** V 



neglecting the powers of b/a higher than the square, we thus obtain R = ^"" ( 1 ., ), 



TTU~ \ 4u"/ 



which is but slightly different from the approximate value paa/irlr calculated on the 

 assumption thnt the elements were all of ;i mean length . In practice, the 

 length of tin- mercury column is assumed as equal to the straight line AB ; also, 

 since the volume of mercury is aourtr, the cross-section is taken as aairlr/AB. 

 Since AB = In sin |a, the calculated resistance- is 4p, sin- ^a'lrb'-ct. Thus the true 



