BIBLIOGRAPHICAL NOTES. 



515 



Form now the determinant A_ 



If A 6 = 0, the lines are in involution. Considering only the figures 1, 2, 3, 4, 5, 

 the determinant A 5 can be formed. If A 6 = and A 5 = 0, the five lines 1, 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 = 0, without any other condition, the five lines 1, 2, 3, 4, 5 

 intersect a single transversal. If A 4 = 0, without any other condition, the lines 

 1, 2, 3, 4 have but one common transversal (Cayley). A determinant can be found 

 which is equal to the square root of A 6 . 



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



This remarkable work, a development of an earlier volume (1844), by the same 

 author, contains much that is of instruction and interest in connection with the 

 present theory. 



A system of n, numerically equal, &quot; Grossen erster Stufe,&quot; of which each pair 

 are &quot;normal,&quot; is discussed on p. 113. A set of co-reciprocal screws is a particular 

 case of this very general conception. 



The &quot; inneres Produkt &quot; of two &quot; Grossen &quot; divided by the product of their 

 numerical values, is the cosine of the angle between the two &quot; Grossen.&quot; If 

 a, 6, c, ... be normal, and if k, I be any two other &quot;Grossen,&quot; then 



cos z kl = cos i ok cos L al + cos ^ bk . cos t bl, + &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 &quot;allgemeine raumliche Grosse zweiter Stufe.&quot; 



The &quot;kombinatorisches Produkt&quot; (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 system of the (n l)th order. 



PLUCKER (J.) On a new geometry of space. 

 1865. 



Phil. Trans., Vol. civ., pp. 725791. 



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 of forces (p. 786). 



This is of importance for our purpose because the linear complex may be also 

 defined with perfect generality as the axes of all the screws of any stated pitch 

 which belong to a 5-system. The relation of the linear geometry to Dynamics is 

 developed in the Theory of Screws. 



HAMILTON (Sir W. R.) Elements of Quaternions. Dublin, 1866. 



In Art. 4 1 6 the equation 2 V (a - y) (3 = 0, is regarded as the single equation of 

 equilibrium when it is satisfied for all values of y, the vector to an arbitrary 

 point C in space. In general if y is not supposed to vary in this arbitrary 

 manner, the equation is that of the central axis. 



332 



