On the Motion of a Plate of Metal on an Inclined Plane. 73 



it is contracted. Two other principal conditions arise in the discus- 

 sion ; one being that in which a part only, and the other that in 

 which the whole of the plate dilates arid contracts. 



In the former the dilatation or contraction is represented by 

 E \ 2 1~ cos <j> 

 2fi(l±\t) sin 0±i)' 

 or by E X 2 t~ cos <p 



2fi(l + \t) sm(0 + i)' 

 according as the plate is fixed at the top or the bottom. 

 In the latter it is represented under the same conditions by 

 g / X f_/f»m(0±,) 1 

 I 2E cos <p J 



or by 



L 2E cos 9 J 



In which formulae — 



a represents the length of the plate. 



fi its weight in lbs. per foot of its length. 



l the inclination of the plane. 



(p the limiting angle of resistance (the angle of friction) between 

 the surface of the plane and of the plate. 



E the modulus of elasticity of the plate. 



X the dilatation or contraction per foot of the length for each 

 variation of 1° of Fahrenheit. 



+ 1° the rise or fall of the temperature in degrees of Fahrenheit, 

 by which the dilatation or contraction of the plate is supposed to be 

 caused. 



In the case in which no part of the plate is fixed, a horizontal line 

 may be taken in it above which it dilates upwards, and below it 

 downwards. The position of this line is determined by the consi- 

 deration that, if the plate be imagined to be cut through along that 

 line, the thrust necessary to push the part above upwards must be 

 equal to that necessary to push the part below downwards. 



In like manner a horizontal line may be found above which the 

 plate contracts downwards and below it, upwards. 



The former neutral line is nearer the top than the bottom, the 

 other nearer the bottom than the top. The one is at the same di- 

 stance from the top as the other is from the bottom. This distance 

 is represented by the formula 



jff sin (0-Q _ 

 2 sin <p cos i 



When the plate is dilated, it is the longer portion which dilates 

 downwards ; and when it is contracted, it is the shorter portion 

 which contracts upwards. The lower end of the plate descends 

 therefore by a given increase of temperature more than it ascends 

 by an equal fall ; and on the whole the plate descends. 



If we suppose the temperature first to be increased by t* s and 

 then diminished by t 3 ° ; then — 





