184 OF THE PRESSURE OF UNMTXABLE FLUIDS OF DIFFERENT DENSITIES 



sin. 20 tan. re (m 1) sin. tan. x = cos.20 -\-(m 1) cos.0 m, 

 and finally, by division, we obtain 



cos.2$ -j- (m 1) cos.0 m 



~l)sin. (141). 



193. We believe that Mr. Barclay's Hydrostatic Quadrant, for 

 finding- the altitude of the heavenly bodies when the horizon is 

 obscure, is founded on principles similar to those propounded in this 

 problem, and expressed in the above equation ; but it would be 

 improper in this place to attempt a delineation of this instrument ; 

 it will therefore suffice, to illustrate the reduction of the above 

 formula, by a numerical example performed according to the direc- 

 tions contained in the following practical rule. 



RULE. From the specific gravity of the heavier fluid, sub- 

 tract unity ; multiply the remainder by the natural cosine of 

 the circular space occupied by each fluid; to the product add 

 the natural cosine of the circular space occupied by both 

 fluids ; then, from the sum subtract the greater specific gravity, 

 and the remainder will be the dividend. 



Again. From the specific gravity of the heavier fluid, sub- 

 tract unity ; multiply the remainder by the natural sine of 

 the circular space occupied by each fluid ; then, to the pro- 

 duct add the natural sine of the circular space occupied by 

 both fluids, and the sum will be the divisor. 



Lastly. Divide the dividend by the divisor, and the quotient 

 will give the natural tangent of a circular arc, which being 

 found in the tables, enables us to assign the position of the 

 fluids when in a state of equilibrium. 



194. EXAMPLE. Suppose that 100 degrees of the inner circum- 

 ference of a circular tube, exhibits equal quantities of mercury and 

 water, whose specific gravities are to one another, very nearly, as 

 14 to 1 ; it is required to assign the position of the fluids, with respect 

 to the vertical diameter of the tube, when they are in a state of equili- 

 brium with each other ; that is, when they excite equal pressures on 

 the plane passing through their common surface ? 



Here we have given mm 14 :0z=:50 , its natural sine and cosine 

 .76604 and .64279 respectively; 20zr 100, its natural sine .98481, 

 and its cosine .17365; therefore, by the rule, we have 



For the dividend, 

 cos.2(H-(w-l)cos.0-?w .17365+13 X. 64279 14r= 5.81738, 



