MOMENT ON A SUBMERGED SOLID OF REVOLUTION 135 
and immersed to a depth f = 2-56; we calculate the part G, of the 
moment for a Froude number 0-5. For a spheroid in a uniform stream at 
a small angle 6 to the axis the moment tending to increase this angle is 
g7pab*c*(k,—k,)8. Comparing these two;moments in this particular case, 
we find that G, would be accounted for in this way by an angle 6 of about 
0-03, which seems not unreasonable. However, this comparison cannot be 
pressed far; it is only intended to indicate a possible physical explanation 
of the negative moment at high speeds. 
7. Consider now any solid of revolution which, so far as axial motion in 
an infinite liquid is concerned, can be specified by a known axial source 
distribution. The part G, of the moment can be obtained at once by the 
method used in the previous sections: but it is not possible, in general, to 
calculate the part G,. Turning to the connexion between G, and R for the 
spheroid given in (23), it is proposed now to use this as a suitable approxi- 
mation for any solid of revolution, and in particular for one of large ratio 
of length to beam. The inertia coefficient k, can be calculated; if M is the 
total moment of the given axial source distribution and V is the volume 
of the solid, we have 47M — (1+h,)V. It is not possible, in general, to 
calculate k,. However, for a long slender solid, k, is small; on the other 
hand, k, approximates to the value unity which it has for the transverse 
motion of a circular cylinder. Thus, in such a case, it is sufficient for a 
fairly close approximation to take 
C= Ren (26) 
where FR is the wave resistance of the submerged solid. The simplest case 
is that of the solid specified by a single source and sink. If mis the strength 
of the source or sink, 2h the distance apart, 2/ the axial length of the solid, 
and 2b the maximum beam, we have 
4mlh = c(l?—h2)2: 4mh = ch?(h?-+b?2)!. (27) 
Taking the axis at depth f, the velocity potential can be written down 
to the same approximation as for the spheroid in (1). The process of 
determining F and the part G, of the moment is the same as before, and 
the details need not be given. Using (27) to express m in terms of the 
dimensions, we obtain 
tn 
R = 2mgpxy,b4(1-+-b2/h2) || {1—cos(2xyh sec O)}e-20 sec*8 see39 dO, (28) 
0 
377 
Gi, = 2mgprg hb*(1+b%/h?) | sin(2e,h sec Be 26/50" sect9 dO, (29) 
0 
581 
