30 



tan.j x r n (x-a) 

 " 



and then having found the fluent, di- 

 vide it by the expression for the figure. 

 Thus, if the figure is a rectangle, 

 whose sides are equal \o b, and base 

 equal to c ; and if the upper edge is at 

 the surface of the water, so that a = 0, 

 the first fluxional expression becomes 



b 

 2 



whose fluent is 



2 c x a 

 ~~irr~ 



this by b c, the area of the figure, we have 



2 x 3 



3 b 2 



which, for the whole rectangle, when 



x = b, becomes - b. The second flux- 



ional expression becomes 



2 tan. 0cx# 2 :c=0 



because the centre of gravity is in the 

 axis and $ = 0. 



If the figure be a Conic Parabola, 

 whose parameter is p, and vertex at the 

 surface of the water, n = 2^pxm,= 



- b and the area- -be; the fluxional 



5 3 



expression, therefore, for the depth, is 



_ . _ 



2 ^/ P xx 2 x _ 10 V p x ^| ^ 

 3 36 



5 



HYDROSTATICS. 



whose fluent is -X^/pxx s ; 

 Ho 



and h = b d dividi b 1 6 c 



' ? 3 



.. 10 , 



li becomes - b_^/_p or, as 



c 



- c 5 



v P b = g , it becomes - 



and therefore is wholly independent of 

 the breadth of the base, or the value 

 of the parameter. Consequently, if 

 an infinite number of parabolas be 

 dividing drawn through any point as a vertex, 



their common centre of pressure will 

 a j ways be a point 5 _ 7ths of the axis 



distant from the vertex. 



The same calculus may be applied to, 

 and the same proposition shewn to hold 

 of, parabolas of all orders. 



^ e _i y = ^ 



for here n== ^ m = tl b . and 



- 



. _ 



J U4-OP' 9 

 &nd the fluxional expre ssion being 



2 (c + 2)aj + a ic 

 e _ l x (g + }) fc , finding the fluent, 



substituting, and reducing, we have 



e i 2 



-- x b for the depth of the centre 



s -j- 3 



of pressure, an expression wholly inde- 

 pendent of the parameter, or of the 

 breadth of the figure. 



