28 REPORT— 1869. 



ence to the disturbance or resistance of the water, is analogous to the free 

 rolling and sliding, on a smooth plane, of the surface which is the envelope 

 of its i^lanes of flotation, the centre of gravity, the upward pressure of the 

 fluid, and the moment of inertia being supposed to remain unaltered. But 

 although this statement reads simply enough, the expressions for the time 

 and the period, which result from it, are exceedingly complex. An inves- 

 tigation of it, subject to the sole restriction that the transverse section of 

 the surface enveloping the planes of flotation shall be circular, has been 

 given by Canon Moscley in the ' Philosophical Transactions ' for 1850, 

 p. 626, and is reprinted in his ' Engineering and Architecture.' The result- 

 ing expression depends upon a hyperelliptic integral. But we are without 

 evidence as to how far the restriction is fulfilled by ordinary ships ; and we 

 do not find reason for supposing that the variation of the radius of curvature, 

 which is thus taken as constant, has ever been practically investigated. There 

 is, however, no difiiculty in extending the formula to the general case ; but 

 it does not appear that the integration can be effected without introducing 

 restrictions. At any rate the value of the integral has not yet been traced, 

 except for small oscillations, when it reduces to the one previously given. 

 There is a reduction in some particular cases*, and notably in the case of 

 isochronous ships. Professor Eankinef has shown that the condition of 

 isochronism is that the curve of buoyancy should be the second involute of a 

 circle described about the centre of gravity. 



It does not appear that the arithmetical consequences of the variation of 

 the law connecting time and angular velocity in unresisted free rolling have 

 ever been worked out. It would be a very laborious business ; and we shall 

 see by-and-by that it is not the chief problem. 



* Let 7c be the radius of gyration, X the lieight of the metacentre above the centre of 

 buoyancy, IT, and H., the clepfJiK of the centres of gravity and buoyancy — all taken for the 

 upright position. Also let 9 be the inclination and e, the extreme, and p the height of 

 the centre of curvature above the actual plane of flotation. Then Canon Moseley's formula 

 gives for the periodic time of the double oscillation 



9 J -e^ {H,-H,-flX (cos e+cosfy,)} (cos0-cos0i) 



It will be observed that (Hj + p) sin 6 is nothing but the horizontal distance between the 

 centre of gravity of the shi]5, and that of the plane of flotation ; or, in other words, the 

 perpendicular from the centre of gravity on the normal to the flotation-envelope. It 

 seems, at the same time, simpler and more general to use this (which we may call v), 

 instead of considering the curvature. We thus get for the periodic time 



V2 r^'^flv/ F+^^^ ~ 



9 1-e, 



9 \ —e {Hj— H2+i\(cos(y-(-cosejl. |cos 0— cos dX 



Now, if V be constant, that is to say, if the flotation-envelope be the involute of a circle 

 described round the centre of gravity of the sliip, this reduces to a complete elliptic inte- 

 gral of the first kind ; but the solution is not mechanical unless i'=0, or the flotation- 

 envelope reduces to a point. When, moreover, the centres of gravity and buoyancy coin- 

 cide, Hj — H., vanislies, and the integral may be at once transformed to its regular expres- 

 sion by writing sin 6 = sin 0, sin <p. We then get for the periodic time 



Va^ Jo 



VA^' Jo VI — sin^ ©1 sin^ 



The time at any moment is got by integrating from —9, to any value of 9 instead of to 

 -fO,.— C. W. M. 



t See Trans. I. N. A. vol. v. for 1864, p. 34. See also Froude "On Isochronism of 

 Oscillation in Ships," Trans. I.N. A. vol. iv. for 1863, p. 211. 



