176 Proceedings of the Royal Irish Academy. 



that there are thus three possible motions without rotation, having in general 

 three different masses. If such a system were constrained to move with a 

 uniform motion of translation in any direction, the constraints would have 

 to produce a couple Fr^if in order to prevent rotation, i.e. there is a 

 force - Vt^it due to motion tending to cause rotation. It is then easy 

 to see that if f differs little in dii-ection from a principal axis this couple 

 will tend to decrease the angle between f and the principal axis, provided 

 that the root g corresponding to that axis is the greatest root of the cubic. 

 So that there is thus one stable direction of motion, and w^e shall see in 

 the next paragraph that this would mean that an elongated body would 

 tend to set itself with its greatest length in the direction of motion, as 

 is easily seen physically from a consideration of the magnetic force. This 

 result is opposite to that of a corresponding theorem in Hydrodynamics. 

 The analogy to cyclic motion can, as is well known, be made by closed 

 conduction currents in the body or symmetrical electrified fly-wheels 

 which will add the necessary " gyrostatic " terms to the energy function. 

 It remains now to consider the effect of the mass of the particles. If 

 M = ^vii, the linear and angular momenta are, respectively, 



Mr + 5^1 1-^ojo-i 

 and 



or 



and 



X^ir) + xM> 



where Xb X"-^ ^^^ xs ^I'e linear vector functions, ^i and xs being self -conjugate, 

 and x'2 the conjugate of xa- Thus the completed expressions for the momenta 

 will be of the same form as before, so that the results already obtained will 

 still hold. 



4. Miscellaneous Peoperties of the Yectoe Fuxctioxs Inteoduced. 



In this paragraph are collected some miscellaneous properties of the 

 functions ^i, ^2, and ^3. 0i is obviously a self -conjugate function. Its invariant 

 nv' or - ^Sitpii where i, j, Jc are rectangular unit vectors is equal to 



which when multiplied by the square of the velocity of light represents four 

 times the "work necessary to collect the system from a state of infinite 

 diffusion. If the body is isotropic about a point such as a sphere, (pi then 

 becomes a constant equal to one-third of the above value. We can in this 



