June 21, 1900] 



NATURE 



187 



corners of the section. This increased efficiency is due to two 

 causes : — 



( 1 ) The rocking of the ship produces currents in the water, 

 which flow round the corners in the opposite direction to that 

 in which the ship is rolling, thereby increasing the pressure on 

 the bilge-keels. 



(2) The discontinuous motion past the bilge-keels alters the 

 listribution of pressure against the sides of the ship, and the 



differences of pressure thus produced have a moment always 

 tending to retard the rolling motion. 



1 he effect of stream-line motions. — Consider a cylinder 

 of section, such as represented in Fig. i, rotating in fluid about 



an axis through its centre O. It is known that the fluid dis- 

 placed by the motion of the cylinder will flow past its protrud- 

 ing corners B, D in the opposite directions to that in which the 

 cylinder is moving ; while at the points A, c, E the fluid will be 

 moving in the same direction as the cylinder. Hence if a small 

 lamina representing a bilge-keel be placed at B or D, it will 

 encounter a current of liquid flowing in the opposite direction 

 to that in which it is moving, and the pressure on the lamina 

 will be correspondingly increased. 



I made several calculations to form some estimate of the 

 increases produced by these counter-currents on the resistance 

 experienced by a suitably placed bilge-keel, assuming the 

 resistance to vary as the square of the relative velocity. 

 Taking certain sections more or less approximating to the form 

 of a square with rounded corners, in a section where the great- 

 est radius o B exceeded the least radius o a by 13 per cent., the 

 resistance was increased, owing to this cause, by about 36 per 

 cent. ; while in the section actually represented in Fig. I, OB 

 exceeded o A by 21 per cent., and the resistance on a lamina at B 

 came out 67 per cent, greater than it would be if the lamina had 

 only to encounter the relative velocity of the fluid due to its 

 own motion. 



In these cases the fluid was supposed to be indefinitely ex- 

 tended. To estimate the influence of surface-conditions 7t//Mo«/ 



taking account of waves, I next considered the case of a cylinder 

 partially immersed in a liquid bounded by a rigid horizontal plane, 

 the cylinder itself rocking about an axis in the plane of the 

 surface. The form chosen for the section of the submerged 

 portion was roughly suggested by a diagram of the ship 

 Revenge, and the counter-currents past the protruding corners 

 were found to be sufficient to more than double the resistance 

 on a suitably placed bilge-keel. 



The ej^ect of pressure-variations against the sides of the ship. — 

 The resistance on a lamina moving in fluid is due to the pressure 

 being greater on the front than on the back of the lamina, this 

 difference of pressure being the result of the discontinuous char- 

 acter of the fluid motion. When the lamina is attached to a 



NO. 1599, VOL. 62] 



ship as a bilge-keel, this difference of pressure also extends 

 along the sides of the ship, the ])ressure in front of the bilge-keel 

 being greater than behind it. Whether the pressure in front is 

 increased or the pressure behind is decreased, or both of these 

 effects take place simultaneously, is unimportant, for we may 

 on each of these hypotheses represent the effect by an excess of 

 pressure on the portions in front of the bilge-keels as compared 

 with the portions behind. 



Now let Fig. 2 represent a diagrammatic section of a ship 

 approaching to a rectangular section, with bilge-keels at its pro- 

 truding corners, B, d. Then if the ship be rolling about its centre 

 of gravity, O, in the direction of the circular arrow, the greater 

 pressures in front of the bilge-keels will be distributed over the 

 segments A B and c D, and, as indicated by the arrows P and Q, 

 their moments about o will be in the direction opposed to rota- 

 tion. When the ship rolls in the opposite direction, the greater 

 pressures will be onthe segments B c and d e, and their moments 

 will again tend to retard the rolling. In this way the pressures 

 on the sides of the ship will materially assist the bilge-keels in 

 steadying the ship. 



To test the extent to which the pressure on the sides of the 

 ship is likely to be modified by the presence of the bilge-keel, I 

 examined the case of a fluid flowing along a plane, A B, with an 

 edge, B c, projecting at right angles to it (Fig. 3). If from B there 

 be measured off on b a a length = -927 B c approximately, the 

 thrust on this portion of B A is equal to the thrust on B c, and the 

 average pressure on this portion is therefore a little greater than 

 the average pressure on B c. If, again, we measure oft" on B A a 

 length = 2*042 BC, the thrust on this portion is equal to twice 

 that on B c. Speaking in general terms, we may say that the 



V. 



Fig. 3. 



pressure is greater along b a than along B c, and that it 

 does not fall off rapidly along the arm v, A. If the arm B a, 

 instead of being straight, bends away from B C like the curved 

 side of a ship, we should expect the pressure on it to be even 

 greater than in the case considered. These considerations led 

 me to consider the results of supposing the bilge-keels to 

 produce on the segments in front of them (a b and c d of^ 

 Fig. 2) uniform increases of pressure equal in intensity to 

 the average pressures on the keels themselves. Applying 

 this hypothesis to a ship, the details of which were 

 kindly furnished me by Dr. Elgar, I found that the 

 total retarding moment came out to be about 3*9 times the 

 retarding moment on the bilge-keels alone. This result, taken 

 in conjunction with the previous result that the currents past 

 the sides of the ship may double the pressure on the bilge-keel, 

 shows that there is nothing paradoxical in the supposition that 

 bilge-keels may have eight times the efficiency in extinguishing 

 oscillations that would be inferred from experiments made with 

 a simple paddle moving freely in fluid. 



We learn from Sir William White that in the case of the 

 Sultan the agreement between Mr. Froude's estimate of the 

 resistance to rolling, based on the coefficient of resistance of a 

 lamina, and the experimental facts was very close indeed. But 

 the midship section of the Sultan was very much more nearly 

 circular than that of the other ships experimented on. For a 

 circular section there would be no counter-currents producing 

 increased resistance, nor would the changes of pressure against 

 the sides of the ship possess any retarding moment. 



There are many further points connected with the investigation 

 which want of space prevents us from discussing here. One 

 such point is the fact that the rolling motion is not steady, but 



