BIBLIOGRAPHICAL NOTES. 537 



the special case when two reciprocal screw-systems have a screw in common. I had 

 overlooked this exception ( 96). The existence of n real principal screws of 

 inertia of a rigid body with n degrees of freedom is proved also by Octonions, 

 p. 248, and also the n harmonic screws, p. 248. 



JOLY (C. J.) The Associative Algebra applicable to Hyperspace. Proceedings of 

 the Royal Irish Academy, 3rd Ser., Vol. v., No. 1, pp. 73-123 (1898). 



The algebra considered in the present paper is that where units i lt i. 2 ... i n 

 satisfy equations of the type i a - - 1 and i s i t + i t i s = 0. In this profound memoir 

 there is a discussion of the Theory of Screws in a space of m dimensions. We 

 learn that &quot;when a system compounded from m screws is defined by a linear 

 function (_/), the reciprocal system is denned by the negative of the conjugate 

 of that function {f}&quot; The canonical representation of a screw in Hyperspace 

 is given and the vector equation to the locus which is the analogue of the cylindroid. 

 The following result, p. 106, is of much interest. &quot; Thus in spaces of even order, the 

 general displacement of a body may be effected by rotations of definite amounts in 

 a number of definite hyper-perpendicular planes, one determinate point being held 

 fixed ; in spaces of odd order, a translational displacement must be added to the 

 generalized rotation ; but by proper choice of base-point this displacement may be 

 made perpendicular to all the planes of rotation.&quot; This remark is illustrated by 

 the well-known laws of the displacement of a body in two or three dimensions 

 respectively. 



JOLY (C. J.) Bishop Law s Mathematical Prize Examination in the University 

 of Dublin, Michaelmas, 1898. 



Many of Sir William Hamilton s discoveries in quaternions were first 

 announced in questions which he proposed from time to time at the Law Prize 

 examination. This is, so far as I know, the only examination in which quaternion 

 problems are still habitually proposed. Professor C. J. Joly in the Law Prize 

 paper for 1898 has given the following questions containing applications of Quater 

 nions to the Theory of Screws. 



(a) The origin being taken as base-point, let //, and A denote the couple and 

 the force of any wrench, then the transformation 



p=?.x=,sr.A+r.A 



A A A 



contains Poinsot s theorem of the Central Moment. 



is the vector equation of a ruled surface (the cylindroid) formed by the central 

 axes of wrenches compounded from two given wrenches (//,,, AJ) and (&amp;lt;u 2 , A.,). 



(c) The form of this equation shows that the locus of the feet of per 

 pendiculars dropped from an arbitrary point on the generators of a cylindroid is a 

 conic section. 



(d) If (M, A) is any wrench compounded from three given wrenches (/^, Aj), 

 (fji.,, A 2 ), and (//..,, A ;! ), the couple of this wrench is a determinate linear vector 

 function of the force, or // &amp;lt;/&amp;gt;A, and the function adequately defines this three- 

 system of wrenclves. 



