PRESIDENTIAL ADDRESS. 473 



case of resisted rolling each increase of extreme angle of roll causes an increase in tha 

 work done upon the resistance of the ship, and when the increase in work done in 

 increasing the angle of heel by each passage of the wave equals the work done in 

 increasing the resistance incurred in swinging through this greater angle, then we 

 shall have a balance of condition and a uniform angle of roll. The angle at which 

 this balance takes place depends on the period and maximum slope of the wave 

 and the coefficients of resistance between the ship and the water. For instance, 

 with a maximum wave-slope of 3° and with a ratio of ship to wave-period of 1*1 tho 

 value of the angle of ultimate uniform rolling in the case of H.M.S. Revenge was found 

 to be 13°-9 without bilge-keels and 10°-8 with them. In the case of synchronism 

 of the ship and the wave, the rolling is uniform always and reaches a maximum of 

 41°- 1 without and 14°85 with bilge-keels. The nearer the wave and ship are to 

 synchronism, the larger is the maximum inclination reached before uniform rolling 

 sets in and during uniform rolling. Resistance is of much more importance in the 

 case of synchronism. If the ratio of ship to wave- period be 1-3, the maximum 

 angle before uniform rolling is reached is 8°-25 without and 6°-6 with bilge- keels, 

 while that due to uniform rolling is 4°-35 without and 4°-24 with. We see, therefore, 

 the important part that the near approach to synchronism plays in creating large 

 angles of roll and the value of bilge- keels in reducing the rolhng in conditions 

 approaching synchronism. When on waves of smaller period, when small angles of 

 roll may be expected, the bilge-keels give but small advantage. The assumption in 

 these cases is that the vessel starts from rest in the upright in the mid-height of the 

 wave, and that the rolling is caused by the assumed uniform swell. The vessel will 

 go through the cyclic change already described and will reach a maximum inclination 

 of not more than double that which she reaches when uniform rolhng has set in. 



A later investigator, Colonel Russo of the Italian Navy, found by experiment 

 that by varying the assumption as to starting condition of the ship, by letting 

 the wave-action begin to operate first when the vessel is upright and at rest on 

 the crest of the wave, the maximum angle before uniform rolhng sets in can be 

 more than four times that due to uniform rolhng if the time of the ship is greater 

 than that of the swell. There is an infinite number of solutions of rolhng amongst 

 waves because there is an infinite number of initial circumstances, but, whatever these 

 may be, the rolhng in a uniform swell will always soon degenerate into a series of 

 uniform forced oscillations in the wave- period. 



From this discovery of Colonel Russo's we see that the region of investigation of 

 possible causes of upsetting is removed from that of uniform rolhng even in a non- 

 synchronous sea. The following table shows for the Revenge with bilge-keels the 

 variation in maximum angle of inclination before and during uniform rolling in terms 

 of the period and length of the swell : — 



Period of swell in seconds . .8 10 12 13' 3 15 17 19 



Length of swell in feet . . . 328 512 738 910 1153 1481 1850 

 Maximum angle in degrees before 



. uniform rolling . . . . 63 8"0 14"7 21'4 17 1 130 ll'O 

 Maximum angle in degrees during 



uniform rolling . 2'5 4"2 12"6 21"4 15'4 IPO 8"7 



The period of free rolling of the Revenge through small angles for a double roll was 

 about 16 seconds. The foregoing shows that the maximum rolling (which occurs 

 at synchronism) took place at a period of swell of 13'3 seconds. The period of roll 

 was less at large than small oscillations. The above figures are for waves varying from 

 Jyth to Jjth of their length in height. The length of wave which corresponds to 

 maximum inclination is 910 feet and height is about one- fiftieth. The maximum 

 wave-slope for such waves is 3°"6. We are in the habit of dealing with waves of 

 one-twentieth of their length in height for strength calculations. Observers have 

 recorded waves in the open ocean of 600 to 800 feet in length and of 30 to 45 feet in 

 height, so that we know that the slope of the waves assumed by Colonel Russo is much 

 less than may be encountered at sea. A wave whose length is twenty times its height 

 has a maximum slope of 9°. Records of waves having a ratio of height to length of 

 as great as one-thirteenth have been published. The maximum slope of wave 

 corresponding to these proportions is 14°. If it is admissible to take much larger 

 angles of wave-slope we may expect to get much larger angles of maximum inclina- 

 tion both before uniform rolhng sets in and when it does. In a case given by Mr. 

 Froude in which the maximum inclination in the Revenge before uniform rolling was 



