APPENDIX I. 187 



invariants of the linear complex with each of six given linear 

 complexes multiplied by certain constants. The six linear com- 

 plexes 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: 



ZEUTHEN (H. G.} Notes sur un systems de co-ordonnees liniaire 

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



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

 on the line decomposed 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 different purpose from 

 that for which the author now quoted has used them. Call the 

 virtual co-efficients of the opposite pairs of edges Z, M, N. If 

 the co-ordinates of a screw with respect to this system be 

 0! . . . a , then the pitch is 



and the virtual co-efficient of the two screws <, 6 is 

 L 



BATTAGLINI (G.) Memoria sulk dinami in involuzione. Atti di 



Napoli IV. (1869). 



The co-ordinates of a dyname are the six forces which 

 acting along the edges of a tetrahedron, are equivalent to the 

 dyname. This memoir investigates the properties of dynames 

 of which the co-ordinates satisfy one or more linear equations. 

 The author shows analytically the existence of two associated 

 systems of dynames such that all the dynames of the first 

 order are correlated to all the dynames of the second. These 

 correspond to what we would call two reciprocal screw com- 

 plexes. 



