Norrbin 



Q = 



<j) - ij; sin 9 



\\} cos 9 sin ^ + Q cos <^ 



4j cos 9 cos «^ - 9 sin <^ 



(4.7) 



The angular velocity components resolved in the inertia frame 



are 



«j) = p+qsin4 ^^^ Q + r cos <^ tan 9 



♦ 



9 = qcos<^-rsin<^ 



ijj = r cos ^sec9 +qsin<^sec9 



(4.8) 



In the special case of motion in a horizontal plane in absence 

 of rolling ajid pitching it is ij; = r» 



In Section VIII an expression will be required for the absolute 

 acceleration of a mass element dm at station P(x,y,z) in a body 

 moving through the water with velocity V. From (4,5) then 



(4.9) 



cLnd by a repeated application of the transformation formula 



u - rv + qw - (q^ + r^)x + (pq - r)y + (rp + q)z 

 V - pw + ru - (r^ + p^)y + (qr - p)z + (pq + r)x 

 w- qu + pv - (p^ + q^)z + (rp - q)x + (qr + p)y 



(M 



abs 



(4.10) 



^_^ In the presence of a homogeneous steady current Vq ^ term 

 AVq is to be added to the right-hand member of Eq, (4.9). In 

 practical applications this current may be as_^umed to take place 

 In planes parallel to the horizontal, so that V Is fully Identified 

 by u^ and v^^ . It Is easy to show that the column matrix for the 

 acceleration In (4. 10) will remain unchanged. To the surface ship 



826 



