1 86 APPENDIX I. 



stants, then the family of linear complexes denoted by 

 x? x* 



r + &C. + j-i-r = O, 



RI - A K 6 - A 



have a common surface of singularities where X is a variable 

 parameter. If the roots A,, &c. be known, we have a set of 

 quasi elliptical co-ordinates for the line x. (Compare with 

 156). 



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. 



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

 Co-ordinaten. Math. Ann., Vol. ii., p. 366-371 (August 4,. 

 1869). 



Let U v , . . U, denote six linear complexes. The moments of a 

 straight line, with its conjugate polars with respect to U^ ... 7,, 

 are, when multiplied by certain constants, the homogeneous 

 co-ordinates of the straight line, and are denoted by x it . . . x t . 

 Arbitrary values of x lt &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 com- 

 plexes is a function of the co-ordinates, or is equal to A sin <f> 

 - (K + K 1 ) cos <, where K and K are the parameters of the 

 linear complexes, A the perpendicular distance, and </> the angle 

 between their principal axes. If this quantity be zero, the 

 two linear complexes are in involution. (The reader will observe 

 that the word involution is here employed in a very different 

 sense to that in which the same word is used by Professor 

 Sylvester.) 



The co-ordinates of a linear complex are the simultaneous 



* This equation expresses that the pitch of the screw denoted by the 

 co-ordinates is zero. 



