538 THE THEORY OF SCREWS. 



(e) Examine the scalar and vector parts of the quaternion &amp;lt;A . A.- 1 . Show 

 that the pitch of any wrench of the system is inversely proportional to the square 

 of that radius of a certain quadric which is parallel to its axis ; also that the locus 

 of feet of perpendiculars drawn from the origin to the central axes of the system 

 is a surface (Steiner s Quartic) containing three double lines intersecting in the 

 origin. 



(/) The screws (/JL, A) and (//. , A ) being reciprocal if &amp;lt;6//A + &amp;lt;S A/^ = 0, show 

 that the screws reciprocal to the system p = &amp;lt;A belong to the system // = - &amp;lt;/&amp;gt; A , 

 or that a linear vector function and the negative of its conjugate determine, 

 respectively, a three-system of screws and its reciprocal three-system. 



Two other theorems communicated to me by Professor Joly may also find a 

 place here. 



If a body receive twists about four screws of a three-system and if the ampli 

 tude of each twist be proportional to the sine of the solid angle determined by the 

 directions of the axes of the three non-corresponding screws, then the body after 

 the last twist will have regained its original position. 



If four wrenches equilibrate and if their axes are generators of the same system 

 of a hyperboloicl, their pitches must be equal. 



WHITEHEAD (A. N.) Universal Algebra, Vol. i., Cambridge (1898), pp. i xxvi, 

 1-586. 



It would be impossible here to describe the scope of this important work, the 

 following parts of which may be specially mentioned in connection with our present 

 subject. 



Book v. Chap. i. treats of systems of forces, in which the inner multiplication 

 and other methods of Grassmann are employed. Here as in many other writings 

 we find the expression Null lines, and it may be remarked that in the language of 

 the Theory of Screws a null line is a screw of zero pitch. 



Chap. II. of the same book contains a valuable discussion on Groups of 

 Systems of Forces. Here we find the great significance of anharmonic ratio in 

 the higher branches of Dynamics well illustrated. 



Chap. in. on Invariants of Groups continues the same theories and is of much 

 interest in connection with the Theory of Screws. 



Chap. iv. discusses among other things the transformation of a quadric into 

 itself, and is thus in close connection with Chap. xxvi. of the present volume. 



Whitehead s book should be specially consulted in the Theory of Metrics, 

 Book vi. The Theory of Forces in Elliptic Space is given in Book vi. Ch. 3, in 

 Hyperbolic Space in Book vi. Chap. 5, and the Kinematics of Non-Euclidian Space 

 of all three kinds in Book vi. Chap. 6. There are also some passages of importance 

 in Statics in Book vn. Chaps. 1 and 2, Book vm. Chap. 4, and on Kinematics in 

 Book vn. Chap. 2 and Book vm. Chap. 4. The methods of Whitehead enable 

 space of any number of dimensions to be dealt with almost as easily as that of 

 3 dimensions. 



STUDY (E.) -Bine neue Darstellung der Krafte der Mechanik durch geometrische 

 Figuren. Berichte iiber die Verhandlungen der koniglich-sachsischen 

 Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-physische 

 Classe, Vol. li., Part IL, pp. 29-67 (1899). 



This paper is to develop a novel geometrical method of studying the problems 

 referred to. 



