1 82 APPENDIX I. 



A, = o and A, = o, the five lines i, 2, 3, 4, 5 are in involution. 

 If all the other minors are zero, the six lines will intersect a 

 single transversal. If A 5 = o, without any other condition, 

 the five lines i, 2, 3, 4, 5 intersect a single transversal. If 

 A 4 = o without any other condition, the lines i, 2, 3, 4 have but 

 one common transversal (Cayley). A determinant can be found 

 which is equal to the square root of A.. This square root is 

 the determinant given in 153. 



GRASSMANN (H.) Die Ausdehnungskhre. Berlin (1862). 



A system of n, numerically equal, " Grossen erster Stufe," of 

 which each pair are " normal," is discussed on p. 113. A set of 

 co-reciprocal screws is a particular case of this very general 

 conception. 



The "inneren Produkte" of two "Grossen" divided by the 

 product of their numerical values, is the cosine of the angle 

 between the two " Grossen." If a, I, c, . . . be normal, and if 

 k, I be any two other " Grossen," then 



cos Lkl = cos Lak costal + cos Lbk. cosZ./, +&c. (p. 139). 



Here we have a very general theory, which includes screw 

 co-ordinates as a particular case. 



In a note on p. 222 the author states that the displacement of 

 a body in space, or a general system of forces, form an " allge- 

 meine raumliche Grosse zweiter Stufe." 



The " kombinatorische Produkt" (p. 41) of n screws will 

 contain as a factor that single function whose evanescence 

 would express that the n screws belonged to a screw com- 

 plex of the (n-ij h order. 



PLUCKER (J.) On a new geometry of space. Phil. Trans., 1865. 

 Vol. 155, pp. 725-791. 



In this paper the linear complex is defined (p. 733). Some 

 applications to optics are made (p. 760) ; the six co-ordinates 

 of a line are considered (p. 774) ; and the applications to the 

 geometry offerees (p. 786). 



