104 Canon Moseley on the Descent of a Solid Body on an 



I. 



Case I. — When the upper end of the plate is fixed. 



Let Ax be a finite increment of x at the temperature T. 



When T became T + £j, Aa* would become l^Ax if nothing 

 were opposed to its dilatation. To dilate, it must, however, 

 thrust the portion P B of the plate (fig. 1) down the plane; and 

 the resistance to this displacement is represented by 



Suppose Ax first to have dilated freely, so as to have become 

 ! } Ax, and then to have sustained a thrust equal to the above- 

 mentioned resistance, and thereby to have been brought back to 

 the dimensions it would have had if it had experienced the re- 

 sistance from the first. 



Let ^(^Ax) represent the compression of the element caused 

 by this thrust. Per foot of the length of the plate this compres- 

 sion is represented by 



B^Ax ) 

 l^Ax' 



But if the same law which holds in respect to small compressions 

 held in respect to all, however great, E lbs. per square inch of 

 section applied as a thrust would, per foot of the length of the 

 plate, produce a compression of one foot ; therefore 



13(7, Aa?) 



l Y . Ax 



will produce a compression of 



Sjl.Ax) 

 /, . Ax 



per foot of length, per square inch of section. But the resistance 



Kw(a — x)f 

 K '~ 



produces also this compression ; 



, ES^Aff) _ Kiu(a-x)f 

 ' '* /j.Aar " K ' 



But 



g(/ x A.2?)= ^'Z 1 {a-x)Ax. 

 A.r^/jAa — 8(/,A*) j 



