612 Dr. C. Chree : Applications of 



there is appreciable shortening of one part of the upper 

 surface and appreciable lengthening of another. Even in the 

 bar C when ajl = ^1/3 we have in the central 2a? 1? or 39*7 cm., 

 a shortening of 1'25/jl. 



If our object is to ensure that the numerically greater of 

 the two changes, contraction of the central portion and 

 extension of the terminal portions, shall be a minimum, then 

 we must select not \/l/3 but 0*5 as the value of ajl. For 

 ajl— 0*5 we have x\ = lj?> ; thus, if the bar be supposed divided 

 into three equal portions, the central portion has its upper 

 surface all under contraction, while the two end portions have 

 their upper surfaces all under extension. Further, the total 

 shortening of the upper fibre in the central portion is exactly 

 equal to the total lengthening of the upper fibre in either of 

 the end portions. 



The mathematical equivalent to this physical property 

 is the fact that for the value 0*5 of ajl the slope at the 

 end is exactly equal but opposite to the slope at the point 

 #=Z/3. 



Another interesting fact is that when a/l=0'5 the point 

 other than the centre where the tangent is horizontal is- 

 given by 



^=i/V3: 



§ 35. As illustrating some of the other results, we may 

 take the case of the Kew standard yard. The actual yard 

 length refers to the distance between the centres of two gold 

 plugs let into the bar so that they lie flush with the upper 

 surface. These plugs are not at exactly equal distances from 

 their respective ends, but the difference in the distances may 

 be neglected for our present purpose ; so that we may treat 

 the distance between the plug centres as representing a central 

 portion 91*44 cm. long in a bar 104*52 cm. long. To keep 

 this length unstretched we must select for ajl a value such 

 that a? 3 in (75) is 45*72, while I is 52*26. The answer to thi& 

 problem will be found to be 



a/£=0*5768, or a=30*14 cm. 



Again, the standard yard has its upper surface graduated 

 into 40 inches, the inch numbered being approximately 

 1*55 em. from one end of the bar, and the inch numbered 3b' 

 about 11*53 cm. from the other end. To secure that the yard 

 to 36 inches shall be unaffected by stretching, we determine 

 ajl from (85), substituting 1*55 for rj u 11*53 for rj 2 , and 52*26 



