MECHANICS. 25 



cal constructions which we have given W sin. e of the weight, in the direction 



in the text. of the plane, in order to obtain the 



Retaining the notation which we have force which is to be balanced by the 



used in pp. 21, 22, the effect of the element of P' in the direction of the 



friction in resisting either the ascent or plane, and must be subtracted from it 



descent of the weight is in order to obtain the element of P" in 



(W cos. e P sin. x) tan. X. the direction of the plane. Hence we 



This must be added to the element have 



AV sin. <?+(W cos. e P' sin. x) tan. X=P' cos. x. 

 AV sin. e(W cos. e-P" sin. x) tan. X=P" cos. x. 



Multiplying both members of each equation by cos. X, and observing that 

 tan. X cos. X = sin. X, we find 



W (sin. e cos. X+sin. X cos. <) P' sin. x sin. X=P' cos. x cos. X 

 "W (sin. e cos. X sin. X cos. #) + P" sin. x sin. X=P"cos. x cos. X 

 .*. W (sin. e cos. X+sin. X cos. e) = P ' (cos. x cos. X+ sin. x sin. X) 

 W (sin. e cos. X sin. X cos. e) = P" (cos. x cos. X sin. x sin. X) 



But, by trigonometry, sin. e cos." X it sin. X cos. e=sin. (e X) 

 cos. e cos. X ;t sin. e sin. X = cos. (e ^p X) 



Hence we obtain, W sin. (e +X)= P' cos. (x - X) 

 W sin. (e X) = P" cos. (x + X) 



'cos. (# X) 



'cos. 

 which formulae are adapted for computation. 



Let us examine under what conditions the two limiting values P ', P " of the 

 equilibrating power will become equal. If this be the case we must have 



sin, (e + X) _ sin, (e X) 

 cos. (x - X) . cos. (x + X) 



.-. sin. % (* + X) cos. (x + X) = sin. (e X) cos. (x X) " 

 .'. 2 sin. (e + X) cos. (x + X) = 2 sin. (e X) cos. (x X) 



But by trigonometry, 



2 sin. (e + X) cos. (x + X) = sin. (e + x + 2 X) + sin. (e x) 

 2 sin. (e X) cos. (x X) = sin. (e + x 2 X) + sin. (e x) 

 Omitting the common quantity sin. (e x} in 'these equals, we have, 



sin. (e + x + 2 X) = sin. (e + x - 2X) 



Hence the angles within the parentheses must be either equal or supplemental. 

 1st. Suppose them equal, 



e+#+2X= e + a? -2X 



/. X = 

 the case in which there is no friction, and therefore but one value of P. 



2nd. Suppose them supplemental, 



<? + # + 2X = 180 -e-x + 2X 



/. 2e+ 2x= 180; 



or, e + x - 90 



/. x = 900 _ e 



Hence the angle x, which the direction of the power makes with the plane, is 

 equal to the complement of the elevation e. This is the same result as was 

 obtained in (34.) geometrically. 



It is very' easy to shew that the geometrical construction in fig. 12, exhibiting 

 the value of P' and P", might be derived from the formulae for these quantities 

 which we have just found, or, vice versa, that the formulae may be derived from 

 the construction. 



In fig. 1 2, W G is equal to F I, or to \V ; the angle W G H is equal to K F I, 



