OF SUBMARINES BY MARBECS METHOD. 259 



Let GX and GY be the principal axes of inertia, then 



L L L 



j X dm = o, C y dm = o, C xydm = o (2) 



000 



The following abbreviations are adopted: 



L L L L 



\ x'dm = iym, C y'dm = tjm, C (x^ + y')clm= C f dm = i 



00 00 



L L L L 



\ {x^ + y') X dm = C f x dm = i^y m, C {x^ + y')ydm= C p^ydm = i 



'p m 



(3) 



All of these quantities are multiples of m, p is the vector from G to any point on the 

 ring, iy and i^^ are the radii of gyration of the ring respectively about the axes GY and GX 

 and if, is the radius of gyration about an axis through G normal to the plane of the ring,- 

 tp^w being the polar moment of inertia about G. The expressions j^^^m and. ip^w are a 

 sort of product-moments of inertia. 



At any point A there is acting the internal reaction Ra and the bending moment Ma. 

 Ra has the components i?^;c andi^^^,. These unknown quantities are determined as usual 

 from the equations of conditions: 



L L L 



C Mdm = o, C M (yx - y) dm = o, C M {xa - x) dm=o (4) 



00 o 



where x, y are the coordinates of a running point P. 

 At any such point P we have : 

 M=Ma + RAy {xa -x) + Ra:^ (yA " j) - — (xa' -2XaX + x^ + yA''-2yAy +y'') 



M= Ma + RAy {xa -x) + Rax {ja -y)- — {^A^ + p' - ^XaX - 2y Ay) (5) 



Substituting in (4) and carrying out the integration, we find, after suitable transfor- 

 mations : 



Put' -^ = b and 44t = « (8) 



(9) 

 (10) 



