620 
STABILITY IN REGARD TO PITCHING AND ROLLING. 
Also AHi = KG— MG=Hi — (Hi cos ?.) = Hi vers d+X; 
W(AHi^AH 2 )=W(Hi^H 2 ) vers 0-\-wz* ; 
(equation 6 .) U(<^, ??)=W(HiipH,) vers ; .... (11.) 
the sign Ip being taken according as the vessel is of the class represented in fig. 3, 
in which the centre of gravity of the displaced fluid ascends, or of that represented in 
fig. 4, in which it descends. 
If a(i be a vertical prismatic element of the space QOS, whose base is dx dy cos 6, 
and height ysin^, then will w.mg be represented, in respect to that element, by 
I 1 
gy mi&.dx dyQO& 9 .-^y s\iid,OY 9 co& 6 y'^dx dy \ and w% will be represented, in 
respect to the whole space of which Pr^Q is the section, by 
^ [M sin®^ cos 9j^y‘^dx dy, 
1 
or by 2 sin^ ^ cos AI. 
If therefore we represent by <p the value of wz, in respect to the spaces of whieh 
the mixtilinear areas PRr and QS^ are the sections, we have 
wz=^/xl sin^^ cos 9-\-(p. 
But the axis O, about which the moment of inertia of the plane PQ is I, is inclined 
to the principal axes of that plane at the angles ?? and about which principal axes 
the moments of inertia are A and B ; and it has been shown by M. DupiN-p that when 
9 is small the line in which the planes PQ or RS intersect passes through the centre 
of gravity of each ; 
.•. I = A cos^;j+B siiPj? ; 
therefore by equation (11.), 
II(^, ;;)=W(H,ipH 2 ) vers ^+^ia/(A cos^jj+R sin^?7)sin^^cos^-i-9. . . . (12.) 
If 9 be so small that the spaces PrR and Q^S are evanescent in comparison with 
POr and QO^, then, assuming (p=0 and cos 9=1, 
U(^, ;j)=:W(H,^H 2 ) vers ^4-^y-'(A cos^;?+B sin^>j)sin¥, ...... (13.) 
which may be put under the form 
U(^, ??) = |w(Hil|lH2)-i-f-'t'(Acos^;j+Bsin®;})|vers 9. 
Again, since 
sin ^=:sin ^ sin (14.) 
* The sign + is here taken to include the case in which the centre of gravity of the displaced fluid descends. 
See Art. 7. 
t Sur la Stabilite des Corps Fiottauts, p. 32. 
