SECT. X.] 



OF STEAM NAVIGATION. 



295 



The agreement is sufficiently near for practical purposes. 



620. The area of the bottom being 4 17 feet, a ton will sink it an inch. The 



fi.QO 



increase of section by adding a ton to the load is therefore ^ = .57 feet ; and 



the increase of surface 12 feet. Adding each ton must therefore increase the 

 resistance about 3'2 Ibs., at the velocity of 2i miles per hour ; therefore the load 

 in tons multiplied by 3'2, added to 51 Ibs. for the boat, will give the force required 

 to draw it. Thus if the load be 20 tons, the force of traction will be 20 x 3*2 + 51 

 = 115 Ibs. 



621. The forms used for vessels are generally curved surfaces of double 

 curvature. To investigate these we may consider them divided into gores, having 

 their bases at the section, and meeting in a point at the water line. A solution on 

 this supposition is fully sufficient for practical objects. Let r be the radius of 

 curvature of the gore, and c its breadth at the base, a being its distance from the 

 axis. Then the differential of the area of the section occupied by the gore 

 will be, 



ex dx 



and, 



c d x 



2 sin. 3 a + sin. 2 a 



= the resistance from pressure. 



By using the approximate equation, 2 r x = y*, we have 



= ? ^- 



d x = ? ^-, and 2- = sin. a, 

 r r 



hence, 



v z c ( JL_ + _L_ -2 ?L_ ) dy = the differential of the resistance. 



\r* 2 r 3 2 r* a 4r*a/ 



Its fluent is, 



- / JL + _J( _ y 8 .} the direct resistance. 



^5r 8 14r2a 24 r a 



