16 OF THE PRESSURE OF FLUIDS OK PHYSICAL LINES. 



3rz s r, the perpendicular depth of the centre of gravity of the 



line e d ; 



^>:n b a A, the inclination of the line a b to the horizon, 

 0'= d c B, the inclination of c d to the horizon, or to the line A B ; 

 j^nrthe relative pressure upon a b, 

 j/zzthe relative pressure upon cd, 

 and 5 the specific gravity of the fluid. 



Now, because the centre of gravity of a physical straight line is at 

 the middle of its length, we have 



a m \ I, and c r rz J /'; 



therefore, by the principles of Plane Trigonometry, we obtain from 

 the right-angled triangle a m n 



d=% lsiu.(j>, 

 and from the right-angled triangle c r s we get 



= | V sin.0. 



consequently, the general expressions for the relative pressures on the 

 lines a b and c d, according to equation (5) are 



p~ \ r s sin.0, and;/ J l'*s sin. 0', 

 from which, by comparison, we get 



p :p' : : I* sin. : Z' a sin. ^'. 



INF. 1. Hence it appears, that the pressures on the lines, when 

 their directions make different angles of inclination with the horizon, 

 Are directly as the squares of the lengths, and the sines of 

 the inclinations jointly. 



2. Where $=<', that is, when the lines are equally inclined to the 

 horizon, whatever may be the magnitude of the inclination, then 



p:p'::l'-.l"; 



therefore, when the lines are perpendicularly immersed, or when they 

 are equally inclined to the surface of the fluid, with which their 

 upper extremities are supposed to be in contact, 



The pressures which they sustain perpendicular to their 

 lengths, are directly proportional to the squares of those 

 lengths. 



3. Consequently, if two or more lines are similarly situated in the 

 same fluid, the relative pressures can easily be compared ; thus, for 

 example : 



Suppose two physical lines, whose lengths are respectively 36 and 

 56 feet, to be perpendicularly immersed in the same fluid, and having 

 their upper extremities in contact with the surface, or equally depressed 

 below it; then, the pressures sustained by these lines, are to one ano- 

 ther as the numbers 1296 and 3136 ; that is, 



pip':: 36 2 : 56' :: 1296 : 3136. 



