330 OF THE POSITIONS OF EQUILIBRIUM. 



the specific gravity of the fluid on which it floats ; when both these 

 conditions obtain, the body will float permanently in a state of equi- 

 librium. 



Put a zz: A D or BC, one of the sides of the section that contain the 



immersed angle, 



b zz: DC or AB, the other containing side; 

 s zz the specific gravity of the floating solid, 

 s' zz the specific gravity of the supporting fluid, or that on 



which the body floats, 

 x zz Dm, the part of the AD which is immersed under mn the 



plane of floatation, 



y zz DW, the corresponding portion of the side DC ; 

 a' zz the area of the whole rectangular section A BCD, and 

 o"zz the area of the immersed portion mvn. 



Then, since the section of the solid is considered to be uniform, 

 with respect to the axis of motion, throughout the whole of its length, 

 we have 



a"s' = a's. (253). 



But by the principles of mensuration, the area of the whole rectan- 

 gular section A BCD, is expressed by the product of its two sides; 

 that is, 



a' zz a b, 



and the area of the immersed triangle mow, is 



Let these values of a' and a" be substituted instead of them in the 

 equation (253), and we shall have 



abs=\xys'. (254). 



By the property of the right angled triangle, it is 



B D 2 zz A D* 4- A B? > 

 or by putting d to denote the diagonal BD, we get 



from which, by extracting the square root, we obtain 



and by the principles of Plane Trigonometry, it is 



\/a 8 + 6 2 : a : : rad. : COS.ADB ; 



and similarly, by Trigonometry, we have 



1^/0? -j- & 2 : b : : rad. : cos. BDC ; 



