348 On the Force^ Construction, <^c. 



the given inclined position of the ship ; y bisecting the angle 

 made by y and y", thus dividing the area, included by y' and y" 

 and the side of the vessel, into two triangles whose sides oppo- 

 site the radial point coincide almost rigorously with the sides. 

 Putting the angle of heeling r= 2 A and rad. = 1, we shall 

 have the sum of the moments of the two triangles with regard 

 to the radial point, and in direction of y" equal to 



{ (7" + 7' COS- ^) 7" + (7' ^^^' ^ + 7 COS. 2A) 7 } ' sin. A. 



and supposing A = 3^°, we have this expression very nearly 

 approximated to by 



{ (7" + 7') 7" + (7' + 7) 7 }-|-'-0305* 



a very symmetrical expression for the horizontal moment of 

 the areas immersed and emerged at an angle of 7", requiring 

 only the simple measurement of the radial ordinates, and but 

 a very moderate degree of trouble in the arithmetical computa- 

 tion of its value f. 



If the sides between the limits of emersion and immersion be 

 straight in a vertical sense, the above expressions will become 

 still more simple for ascertaining the moments of the areas 

 emerged and immersed, for the formula 



/ ,, . ^ *\7 7" sin. 2 A 



(y/ + ^ COS. 2 A)1^I-- 



o 



will shorten the arithmetical operations in those portions of the 



sides which are plane surfaces. At an angle of heeling of 



7° it becomes very nearly 



(7" + 7) 7 7" X -0203. 



The expressions just given are only particular results, as 



we have before remarked, of a very general principle of calcu- 



* In using thin formula, it is evident the operation will be further simplified 

 by the application of logarithms. 



t The simple areas of immersion and emersion will be expressed by 



2 ' 



a formula in every way calculated to save trouble in finding the solids immersed 

 and emersed, and the positions of their centres of gravity with regard to the length 

 whereby we may determine whether the longitudinal axis of rotatiou remains in the 

 diametral section as the ship heels. 



