OF THE POSITIONS OF EQUILIBRIUM. 331 



therefore, by working out the above analogies, and putting radius 

 equal to unity, we shall have 



COS.ADBIZZ , and cos. BD cur . 



Since gG and FP are parallel, and g? equal to one third of DF ; it 

 follows, that GP is equal to one third of DP; or which is the same 

 thing, DP is equal to three fourths of BD ; that is 

 DPz=|Va 2 -f b\ 



When two sides of a plane triangle are given, together with the 

 angle of their inclination, as is the case in the triangles WIDP and 

 WDP; then, the writers on Trigonometry have demonstrated, that 



7WP 2 ZZDW 2 -f- DP 2 - 2D97Z.DPCOS.ADB, and WP 2 DW 2 -|- DP 2 - 

 2DW.DPCOS.BDC ; 



and these, by the principles of floatation, are equal, hence we get 



Dm 2 - SDTW.DPCOS.ADBUZDW 2 - 2DW.D P COS.B DC. 



Let the analytical expressions of the several quantities Dm, on, DP, 

 cos. ADB and COS.B DC, be substituted in the above equation, and we 

 shall obtain 



*-!:=,- ( 255 ). 



If both sides of the equation (254), be divided by the expression 

 Jars', we shall obtain as follows, viz. 



the square of which, is 



y ~ : 



Now, if these values of?/ and z/ 2 , be respectively substituted instead 

 of them in equation (255), we shall obtain 



3ab*s 



2 __ 



" : : 



and finally, by reduction and transposition, we get 



(256). 



And if we consider the value of s', or the specific gravity of the 

 fluid, to be expressed by unity, as is the case with water ; then the 

 above general equation becomes 



