THE EQUILIBRIUM OF BODIES IN CONTACT. 473 



The supposition a, = = gives the case of an upright Pier with 

 horizontal intersections (Fig. 8), and the equation (18) to the line of 

 resistance becomes 



8{2Psin<t> + ia'{ + 2 4 P/ccosO* , , 



y = — r~ji i (22), 



" ax + 2P cos (t> ^ ^ 



the equation to a rectangular hyperbola, whose axis is by formula (19) 

 inclined at an angle of 45" to the axis of x, whose asymptotes are 

 therefore vertical, and the co-ordinates of whose center C are by for- 

 mulse (20), 



.„ SPsin(^ , , .-_ SPeos* 

 AK = + ha, and KC = . 



a ^ a 



CE being an asymptote to the hyperbola, it is clear that if AK he less 

 than AD, that is, if 



SPsind) . 1 ,, 1 



be less than ia, 



a •' 



or 22Psin(I> be less than a', 



the line of resistance wiU not meet the extrados of the pier however 

 great may be its height. But that if 



22Psin$ be greater than «■, 



it will somewhere cut the extrados ; there is, therefore, in this case, a 

 certain height of the pier beyond which, if it be continued, it will be 

 overthrown. 



This maximum height of the pier is determined by the equation 

 S + Pk cos $ - aSP cos <I> 



^a'-2P sin* 



.(23). 



• This equation may be put under the following form, whence all the circumstances 

 mentioned in the text are apparent, 



( /SPsinO , M / SPcostD) 1_ „, ,„ /2Psin<t> ,l^„ ,^ 



