166 OF THE PRESSURE OF FLUIDS OF VARIABLE DENSITY 



radius EF, through which the chord of maximum pressure is supposed 

 to pass; draw the chord GH, and the radius EG; then, because the 

 chord GII is parallel to the diameter ab y it follows, that GH is bisected 

 in m by the vertical radius EF; consequently, m is the place of the 

 centre of gravity of the chord GH, and Em is its perpendicular depth 

 below AB, the upper surface of the fluid. 



Put r IZTEG, the radius of the semi-circular plane, 



= GF, half the arc subtended by the required chord GH, 

 and x Em, the distance of the chord below the surface of the fluid. 



Then, because by the conditions of the problem, the density of the 

 fluid varies directly as its depth ; it follows, that the pressure on the 

 chord GH varies directly as Gra drawn into Em 2 ; that is, 



where p denotes the pressure upon the chord ; but this, by the condi- 

 tions of the problem, is to be a maximum ; therefore, we have 



x* \/ r 2 # 2 n: a maximum, 

 from which, by equating the fluxion with zero, we get 



or by transposing and expunging the common factors, we obtain 



3z 2 = 2r 2 ; 

 therefore, by division, we have 



z'zn-fr 2 , 

 and finally, by evolution, it becomes 



* = rV?. (130). 



The same result, however, may be otherwise determined ; for by 

 the arithmetic of sines, we have, to radius unity 



Gmzzsin.0, and Emzncos.0 ; 



but in order to accommodate these quantities to the radius r, it is 



Gmmr sin.0, and Em 2 zziiK 2 z=:r 2 cos. 2 ^; 



consequently, by multiplication, we obtain 



and this, by the conditions of the problem, is to be a maximum ; 

 hence we get 



r*sin.0 cos. 2 ^ = a maximum, 

 which being thrown into fluxions, becomes 

 r 3 (0- cos. 3 < 20- sin. 2 ^ cos.^) ; 



