OF THE PRESSURE OF FLUIDS ON PHYSICAL LINES. 13 



point, according to circumstances ; then is mn the depth of the centre 

 of gravity of the line a b, below the surface of the quiescent fluid, and 

 ad, be are respectively the depths of its extremities, b AC being the 

 angle which the direction of the given submerged line makes with the 

 horizontal line A B. 



Put d=ad, the depth of the upper extremity of the given line, 

 ^zrmTi, the depth of the centre of gravity, 

 D in b c, the depth of the lower extremity, 

 I zz a b y the length of the proposed line, 

 p zz the relative pressure upon it as propagated by the fluid, 

 and ^ zz b A c, the angle which the given line makes with the horizon. 

 Through a the upper extremity of the given line, draw ae parallel 

 to AB the surface of the fluid; then is the angle bae equal to the 

 angle b A c, and by the principles of Plane Trigonometry, we have 



a b : b e : : rad. : sin. <f>; 



but be is manifestly equal to be ad; that is, 6ezzD c?, and 

 according to our notation, ab l; hence, the above analogy becomes 



I : (D c?) : : rad. : sin. <p, 

 or by putting radius equal to unity, we get 



This equation enables us to determine the obliquity of the line, 

 when the perpendicular depths of its two extremities are given ; but 

 when it is required to determine the relative pressure from the same 

 data, we have only to observe, that mn the perpendicular depth of 

 the centre of gravity, is equal to half the sum of the depths of the two 

 extremities ; that is, 



consequently, we obtain 



Again, if the angle of inclination and the perpendicular depth of 

 one extremity of the line are given, together with its length, the per- 

 pendicular depth of the other extremity can easily be found ; thus, 

 suppose that da is the given depth, then, by the principles of Plane 

 Trigonometry, we have 



b e zz I sin. ; 



but by addition, we obtain 



b c~b e -\~ e c ; that is, D zz I sin. -f" d ; 



consequently, the perpendicular depth of the centre of gravity, is 



z= | I sin. <f> -\- d; 

 and the relative pressure becomes 

 />zz J I* sin. p Id. 



