368 "Proceedings of the Royal Irish Academy. 



Trrenches describe surfaces which are in general of the fourth degree. 

 These surfaces may be obtained from right hyperboloids by projecting 

 a circular section into an ellipse, and by transporting the generators 

 of one system parallel to themselves so as to pass through correspond- 

 ing points on the ellipse. They become right circular cylinders when 

 the axes are parallel, and hyperboloids of revolution when the axes 

 are of equal pitch — a possible case. 



Again, we may write the equation of an axis of a wrench in the 

 form 



p = y + Ct, where t = 5- (2/3)"^ g~'^ = qroq'^ 



is the reciprocal of the vector representing the resultant force, and 



where - C is the virial of the system at the extremity of the vector p. 



If we replace y in (A) by p - Ct, or rather by p - C^r^q''^, we have 



fq- pq+ Cqrt, = cq. (C) 



This equation connects the quaternion q specifying the rotation, 

 the vector p to an arbitrary point, the virial at that point, - C, and 

 the pitch, c, of the resultant wrenches whose axes pass through the 

 point. 



If, for a given value of p, we form the latent quartic, we have the 

 relation between pitch and virial of each of the force systems whose 

 axes pass through the point. If we regard q and C as unknown, we 

 can, from the latent quartic, determine four values of C, and corre- 

 sponding to these four different rotations, so that the resultants of the 

 four corresponding force systems are wrenches of given pitch whose 

 axes pass through the given point. 



For proper choice of origin, and proper choice of the initial position 

 of the force system, the function / becomes greatly simplified. In its 

 simplest form, 



fq = eSq + 4i Vq, 



where the function is self- conjugate, and one of its latent roots is 

 - e, and the others are equal and opposite — say ± e'. In this case, 



(0 + ^)2/3 = 0. 



"WTien this simplification is introduced, it is apparent that the latent 

 quartics are peculiarly simple, being in fact quadratics in c*. Indeed, 

 the latent quartic of (A) and of (B) is 



{T'y + c'- e") {T'y + c^ - e'^) + T'{cj, + e)y = 0. 



The quadrics used in determining the rotations which produce 



