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tangular) is fixed is first discussed ; and then that in which the lower 

 edge is fixed. Each of these cases is considered suhject to the con- 

 dition of friction ; first, when the plate is dilated, and secondly, when 

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

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

 which the whole of the plate dilates and contracts. 



In the former the dilatation or contraction is represented by 

 E X a t z cos ft 



M) sin (ftt) 

 or by 



Ex 2 ; 2 cos ft 



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

 In the latter it is represented under the same conditions by 



cos ft 

 or by 



r x ^sin(ft + t ) 1 

 I 2E cos ft J 



cos ft 



In which formulae 



a represents the length of the plate. 



fj. its weight in Ibs. per foot of its length. 



i the inclination of the plane. 



ft 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. 



