534 THE THEORY OF SCREWS. 



given pair. In this way the existence of two set.s of finite motions, each 

 individual motion of which leaves unchanged each of a doubly-infinite number of 

 straight lines is demonstrated (the right- and left- vectors). It is also shown that, 

 when a right- (left-) vector is represented as the resultant of two rotations 

 through two right angles, the axes of the two rotations are left- (right-) parallels. 

 Hence from the original construction the resultant of two right- (left-) vectors is 

 again a right- (left-) vector. 



Lastly it is shown, still synthetically, that every right-vector is permutable 

 with every left- vector, thus proving in a different way what is given in 427 of 

 this volume. See also p. 526. It is also shown that the laws according to which 

 right- (left-) vectors combine together are the same as those by which finite rotations 

 round a fixed point combine. 



The remainder of II. is concerned with the determination of all distinct 

 types of &quot;continuous groups&quot; of motions in elliptic space. 



III. gives the application of the construction to hyperbolic space. Here 

 attention is first directed to a type of displacement which leaves no finite point or 

 line undisplaced. It is also shown that no displacement in hyperbolic space 

 can leave more than one real line unchanged. This fact, combined with the 

 properties of the previously mentioned special type of displacements, is then used to 

 determine all the distinct types of continuous groups of motion in hyperbolic space. 



JOLY (C. J.) The Theory of Linear Vector Functions. Transactions of the Royal 

 Irish Academy, Vol. xxx., pp. 597-647 (1894). 



In this memoir the close connexion between the quaternion theory of linear 

 vector functions and the Theory of Screws is developed. &quot; The axes of the screws 

 of the resultants of any wrenches acting on three given screws belong therefore to 

 one of the congruencies of lines treated of in the present paper, and every 

 geometrical relation described in it may be applied to problems in Rational 

 Mechanics.&quot; A remarkable quintic surface is discovered which under certain 

 conditions degrades into the cylindroid. At the close of the memoir the linear 

 vector functions expressing screw-systems of the third, fourth and fifth orders 

 are discussed. 



BALL (R. S.) The Theory of Pitch Invariants and the Theorij of Chiastic 

 Hotnography. Tenth Memoir. Transactions of the Royal Irish Academy, 

 Vol. xxx., pp. 559-586 (1894). 



It is shown that if a x ... a 6 be the six co-ordinates of a screw a, while 7i 1} ... h 6 

 are the angles which a makes with the six co-reciprocal screws of reference, then 

 expressions of the form 



a x cos Aj + . . . + a 6 cos A 6 



are invariants in the sense that they are unaltered for every screw on the same 

 ray as a. 



If k lt ... k 6 be the similar angles for any other screw, then 



cos AJ cos & x cos A 2 cos & 2 cos h R cos k 6 



+ + . . . + = U, 



Pi P-2 P 6 



where i\ ... p 6 are the pitches of the screws of reference. 



If two instantaneous screws a and ft and the corresponding impulsive screws 

 77 and are so related that a is reciprocal to , then ft must be reciprocal to 77. 

 This clearly implied that there must in all cases be some relation between the 

 virtual coefficients w a and w^^. The relation is here shown to be 



COS (077) 



