162 OF THE POWER OF STEAM, [SECT. iv. 



cylinder, excepting the small difference arising from the friction not increasing 

 exactly in the same proportion as the square of the diameter ; and this difference 

 is so small in ordinary proportions, that we may safely neglect it. 



328. The only other circumstance which renders it necessary to attend to the 

 proportions of a cylinder is the quantity of cooling surface to which the steam is 

 exposed during its action. This surface ought to be the least possible ; for its 

 effect in condensing, and therefore of destroying the power of the steam, is con- 

 siderable. (See art. 156.) 



The quantity of surface consists of one end of the cylinder, one side of the 

 piston, and the concave surface of the cylinder ; but the latter is only gradually 

 brought into contact with the steam during the stroke, and its effect, therefore, 

 only equivalent to about half the effect of an equal surface bounding the steam 

 during the whole of the stroke. Now the power of an engine is greatest when the 

 effect of a given quantity of steam is the most possible ; hence the question is 

 to find the least surface capable of confining a given quantity of steam, during 

 its action. 



329. When the length of the stroke is twice the diameter of the cylinder, a 

 given quantity of steam is bounded by the least possible quantity of surface during 

 its action in the cylinder ; 1 hence I conclude it is the best proportion for the 

 cylinder of a steam engine, except when the space for the engine limits the length 

 of the stroke : and the same conclusion applies to both atmospheric and steam 

 pressure engines. 



1 Let the diameter of the cylinder be x, its length /, its capacity C, and ir = 3-1416. Then, 



C = * x the sum of the areas of the bases = ^L. ; and the area of half the concave surface 

 4 2 



=- - = - ; hence the whole surface of the steam exposed to cooling surfaces during its 



action is 



* 9 C 



~2~ ~ 5 



and this surface is to be a minimum, which is determined by taking its differential and making it 

 equal to zero. That is, 



, 2C<r 

 x dx - = o ; 



whence, 



and substituting for C its preceding value, we have 2 x = /; or a cylinder is of the best proportion 

 when its length is twice its diameter. 



