180 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



There are therefore three permanent screw-motions such that the correspond 

 ing impulsive wrench in each case reduces to a couple only. The axes of 

 these three screws are mutually at right angles, but do not in general 

 intersect. 



It may now be shewn that in all cases where the impulse reduces to a 

 couple only, the motion can be completely determined. It is convenient, 

 retaining the same directions of the axes as before, to change the origin. 

 Now the origin may be transferred to any point (#, ?/, z) by writing 



u + ry-qz, v+pz-rx, w + qx-py, 



for u, v, w respectively. The coefficient of vr in the expression for the kinetic 

 energy, Art. 118 (7), becomes -J3&+M&quot;, that of wq becomes Cx + N , and so 

 on. Hence if we take 



M&quot; N . N 



the coefficients in the transformed expression for 2 T will satisfy the relations 

 M&quot;/B=N /C, N/C=L&quot;/A, L /A = M/B ................ (ix). 



If we denote the values of these pairs of equal quantities by a, /3, y re 

 spectively, the formulae (ii) may now be written 



d d* d* 



u=-j-, v=-- r -. w= =- ..................... (x), 



dp dq dr 



where 2* (p, q, r) = j p 2 + -^ f + -^ r 2 + Zaqr + Z$rp + Zypq ...... (xi). 



The motion of the body at any instant may be conceived as made up of two 

 parts ; viz. a motion of translation equal to that of the origin, and one of 

 rotation about an instantaneous axis passing through the origin. Since 

 } rj, =Q the latter part is to be determined by the equations 



d\ da dv . 



-dr r -i&quot; di = p&quot;~ r ^ arf-** 



which express that the vector (X, /u, v) is constant in magnitude and has a fixed 

 direction in space. Substituting from (iii), 



d de_ de de 



dt dp dq dr 



d de de de 



d^de_ de_ de 

 dt dr~ q dp P dq 



(xii). 



These are identical in form with the equations of motion of a rigid body 

 about a fixed point, so that we may make use of Poinsot s well-known solution 

 of the latter problem. The angular motion of the body is therefore obtained 

 by making the ellipsoid (vii), which is fixed in the body, roll on the plane 



\x -\- p.y + vz = const. , 



