596 Dr. C. Chree: Applications of 



the numerals refer to the dimensions, and are in millimetres. 

 A refers to a standard yard, B to the international prototype 

 metres of the so-called X-section, to a working standard 

 belonging to the Bureau International, D and E to deflexion- 

 bars used in magnetometers. A and C are divided on their 

 upper surfaces, B on the neutral plane, D on one of the 

 vertical faces ; E has holes on its upper surface into which a 

 plug fits. B and C are copied from vol. vii. of the Bureau 

 International's Travaux et Mdmoires. Their shape is de- 

 vised partly with the object of facilitating equalization of 

 temperature throughout the bar. Most modern standards are 

 supported not over the whole lower surface but on two 

 symmetrical rollers, or on three points, one at one end of the 

 bar, and two — in the same cross section — at the other end. 

 This mode of support is intended to promote uniformity of 

 temperature, the bar being surrounded by liquid, and to 

 remove the uncertainty as to the distribution of surface- 

 pressure when a bar rests on an ordinary table, and not on 

 -an ideal smooth plane *. 



An exact solution of the elastic problem presented by a 

 heavy bar supported on points or rollers has not yet been 

 obtained even for a rectangular section, and the best thing 

 to be done is probably to apply the ordinary approximate 

 Bernoulli-Euler solution. From the researches of St. Venant, 

 Pearson, and others we have grounds to believe that for bars 

 like standard yards and metres, whose length is a large 

 multiple of their greatest lateral dimension, the Bernoulli- 

 Euler solution represents a high degree of accuracy except 

 perhaps in the immediate vicinity of the supports. Actual 

 observations on standards of the types B and (J by Broch and 

 Benoit (Trav. et Mem. 1. c.) seem to bear this out. The 

 Bernoulli-Euler method of solution is so well known that it 

 is unnecessary to describe it, and the results which I am now 

 about to give are deducible from the solution without much 

 difficulty. Some of them have, I find, been given by Airy 

 and Broch, /. c, and possibly this remark applies to more 

 than I am aware of ; but I do not think they are generally 

 known. 



Applications of Bernoulli-Euler Method. 



§ 21. In figs. 2 to 6, OB represents the half of a bar of 

 uniform section co and total length 21 supported symme- 

 trically at two points in a horizontal plane at a distance 2a 



* See Airy (Phil. Trans, for 1857). 



