Elastic Solids to Metrology. 603 



These results have a real significance only when ajl 

 exceeds 0*5505. 



We can also in cases (ii.), (iii.), and (iv.), have unchange- 

 ably of length in an asymmetrical portion which includes 

 the centre. 



Supposing the abscissae of the two extremities f 2 and — £ 2 , 

 we require the values of dyjdx answering to +fi and +£2 

 to be equal and opposite in sign. The points answering to 

 •+- f 1 and + f 2 must thus lie on opposite sides of the " summit,' 3 

 where the slope vanishes. 



Supposing J 2 larger than f 1? then in case (ii.) f x must be 

 less than a, while in case (iii.) f. 2 must exceed a ; in case (iv.) 

 both f , and f 2 must exceed a. The position of the support 

 is easily found ; thus when f x and f 2 are both to exceed a 

 we find 



a/l= i/{2-(W0 3 -(W0 s }/6, . . . (85) 

 where % (or I — &) and -n 2 (or I— £ 2 ) are the distances of the 

 ends of the asymmetrical portion from the two ends of 

 ihe bar. 



Bar under additional weight. 

 § 26. In dealing with the bending of a bar under its own 

 weight we have assumed it homogeneous, and of uniform 

 section, and symmetrically supported. By having the 

 supports unsymmetrical one could obtain un changeability of 

 length in an infinite variety of ways, but to discuss this fully 

 would require more space than the practical importance of 

 :he subject seems to warrant. Before passing, however, to 

 :he practical applications, there is one point requiring notice. 

 A bar not infrequently has to carry a weight additional to 

 its own. For instance, the deflexion-bar of a magnetometer 

 supports a magnet and its carriage on one arm, and unless 

 clamped it carries, or at all events should carry, a counter- 

 poise on the other arm. It is thus desirable to investigate 

 the additional bending due to this cause. We shall suppose 

 symmetry as before, the weight carried on one side of the 

 centre being W, and its distance from the centre c. As 

 before, represents the centre, A the support, B the end of 

 the rod ; also C denotes the position of the weight. The 

 notation used is the same as in § 21. The results given by 

 the Bernoulli-Euler method are as follows, it being assumed 

 that lies between A and B : 

 Between and A, 



y/(^ J ) = J/(*)=3=(W/E «')(<!-«). • (86) 



