518 THE THEORY OF SCREWS. 



have a common surface of singularities where A. is a variable parameter. If the 

 roots Aj, &c. be known, we have a set of quasi elliptic co-ordinates for the line x. 

 (Compare 234.) 



It is in this memoir that we find the enunciation of the remarkable geometrical 

 principle which, when transformed into the language and conceptions of the Theory 

 of Screws, asserts the existence of one screw reciprocal to five given screws. ( 25.) 



KLEIN (F.) Die allgemeine lineare Transformation der Linien Coordinaten. 

 Math. Ann.; Vol. n., pp. 366-371 (August 4, 1869). 



Let /j, ... U 6 denote six linear complexes. The moments of a straight line, 

 with its conjugate polars with respect to C/j, ... U 6 , are, when multiplied by certain 

 constants, the homogeneous co-ordinates of the straight line, and are denoted by 

 x ly ... x 6 . Arbitrary values of a^, &c., do not denote a straight line, unless a 

 homogeneous function of the second degree vanishes*. If this condition be not 

 satisfied, then a linear complex is defined by the co-ordinates, and the function is 

 called the invariant of the linear complex. The simultaneous invariant of two 

 linear complexes is a function of the co-ordinates, and is equal to 



A sin &amp;lt; - (K + K ] cos &amp;lt;/&amp;gt;, 



where K and K are the parameters of the linear complexes, A the perpendicular 

 distance, and &amp;lt; the angle between their principal axes. 



The co-ordinates of a linear complex are the simultaneous invariants of the 

 linear complex with each of six given linear complexes multiplied by certain 

 constants. The six linear complexes can be chosen so that each one is in involution 

 with the remaining five. The reader will easily perceive the equivalent theorems 

 in the Theory of Screws. K and A r are the pitches, and the simultaneous invariant 

 is merely double the virtual coefficient with its sign changed. 



ZEUTHEN (H. G.) Notes sur un systeme de coordonnees lineaires dans I espace. 

 Math. Ann. ; Vol. i., pp. 432-454 (1869). 



The co-ordinates of a line are the components of a unit force on the line decom 

 posed along the six edges of a tetrahedron. These co-ordinates must satisfy one 

 condition, which expresses that six forces along the edges of a tetrahedron have a 

 single resultant force. The author makes applications to the theory of the linear 

 complex. 



Regarding the six edges as screws of zero pitch, they are not co-reciprocal. It 

 may, however, be of interest to show how these co-ordinates may be used for a 

 purpose different from that for which the author now quoted has used them. Let 

 the virtual coefficients of the opposite pairs of edges be L, M, N. If the co 

 ordinates of a screw with respect to this system be O l ... G , then the pitch is 



(LBA + M0 3 0, + N6&\ 



and the virtual coefficient of the two screws &amp;lt;/&amp;gt;, is 

 \L (0^, + 0^) 



BATTAGLINI (G.) Sulle serie die sistemi diforze. Napoli Rendicoiito, viii., 1869, 

 pp. 87-94. Giornale di Matemat, x., 1872, pp. 133-140. 



This memoir deserves special notice in the history of the subject inasmuch as 

 already remarked in 13 it contains the earliest announcement of the dynamical 

 significance of the cylindroid. Battaglini here shows that the cylindroid is the 

 locus of the screws on which lie the wrenches produced by the composition of two 

 variable forces on two fixed directions. See also p. 520. 



* This equation expresses that the pitch of the screw denoted by the co-ordinates is zero. 



