664 Transactions. — Miscellaneous. 



the perpendicular drawn upwards from it will intersect the 

 centre-line of the upright position at a higher point, and thus 

 make the metacentric height greater. At the same time, it 

 will be seen that this perpendicular is further to leeward of 

 the line falling from the ship's centre of gravity, and con- 

 sequently the righting-iever is longer. Now, if I can show 

 that there is in all cases a proportionate increase of these 

 two measurements I have proved the value of the meta- 

 centric height as a comparative measure of stability. The 

 proof is as follows : If we careen several vessels, whose relative 

 stability we wish to know, to any given angle, ascertain their 

 centres of gravity, and calculate their metacentric heights, 

 and then construct a triangle on the section of each ship from 

 its own ascertained measurements, we shall find that the 

 three angles of each triangle are similar in all cases. The 

 upper angle is that of the inclination ; that between the 

 righting-lever and the perpendicular to the metacentre is 

 a right-angle ; while the third is equal to a right-angle miniLS 

 the angle of inclination. We have therefore a series of 

 triangles, all of whose angles are similar. In such cases it 

 follows, also, that their sides must be in the same proportion 

 throughout. This being so, the metacentric height and 

 righting-lever, being two of them, are, as above stated, 

 always in the same proportion to each other. 



Let us now return, from this further explanation of its 

 important relationship, to the consideration of the righting- 

 lever. It will be evident that, as we can calculate this 

 measm-e for one angle, we can do it for as many as may seem 

 desirable, and thus obtain the range of stability. The large- 

 ness or smallness of this range depends entirely upon the 

 relative form of the vessel's sides immediately above and below 

 the water-level section. 



Before branching off just now into the division of the ques- 

 tion of stability into two phases, I had shown, in the case 

 supposed, that the act of careening had so altered the under- 

 water form of the vessel that the centre of buoyancy had 

 been removed from its original position towards the en- 

 larged or lee side of the section. It has since been shown 

 that this results in the lengthening of the righting-lever, 

 and consequently in a gain of stability. Now, let us 

 assume that the length of this lever has been ascer- 

 tained for a vessel at 10^ 20^ 30°, 40°, 50°, &c., of inclination. 

 Having obtained these lengths, set them up as ordinates from 

 a base-line previously divided into the respective angles of 

 heel. Through the points so obtained run a line, which will 

 be that referred to by Mr. Barnaby in 1871 as a curve of 

 stability, then for the first tmie calculated for an actual ship. 

 Having traced this curve, we can read off upon it the length of 



