OF THE EQUILIBRIUM OF FLOATATION. 269 



which the body floats, with the plane of floatation originally passing 

 through EF, but which, on the addition or subtraction of the weight rv, 

 ascends to i K or descends to a b. 



Put p zr the parameter of the generating curve, 



S ~ ec, the distance of the vertex below the surface of the 



fluid at first, 

 m r the magnitude or solidity of the paraboloidal frustum, of 



which IKFE is a section, 



#/:=r the magnitude of the frustum whose section is EF^a ; 

 x rn de, the descent of the body in consequence of the addi- 



tion of the weight rv, 



x' rr ec, the corresponding ascent in the case of subtraction, 

 and s rzi the specific gravity of the fluid. 



Then we have crf = 5 -j- x, and c c rr 8 otf, and according to the 

 writers on mensuration, we have 



and in a similar manner, we obtain 



m' = 1.5708p(28* f *'2) ; 



and since the weight of any body is expressed by its magnitude drawn 

 into its specific gravity, it follows, that the weight of a quantity of 

 fluid equal respectively to m and m 1 , are 



THSIZ: 1.5708;?s(23;r + x 2 ), andm's 1.5708ps(fcc' x' 3 ). 

 Now, these according to the conditions of the problem, are respec- 

 tively equal to the disturbing weight ; hence we have in the case of 

 addition, 



m = 1 .57Q8ps(Z$x + * 2 ), (208). 



and in the case of subtraction, it is 



tv= 1.5708^5(2^' *' 8 ). (209). 



330. The equations which we have just obtained, are precisely the 

 same as would arise, by taking the fluent of the expression in equation 

 (206) ; it therefore appears, that although the fluxional notation is the 

 most convenient for expressing the general result, yet in point of sim- 

 plicity as regards symmetrical bodies, there is little advantage to be 

 derived from its adoption. 



Suppose that in the first instance, the equilibrium is destroyed by 

 the addition of the weight rv, and let it be required to determine how 

 far the body will descend in consequence of the addition. 



Equation (208) involves this condition ; consequently, if both sides 

 be divided by the expression 1 .5708^5, we shall obtain 



