( 824 ) 



VI. We shall now denote still, by means of a few words, in which 

 wav we can arrive at an extension of the screw -theory of Ball by 



the application of the principle of exchange of space-element to the 



10 

 equations 2 ai^=: 0. 



l 



By interpreting' this equation either 



1 st . as condition of united position of a point Xand an Sp s £?in Sp 9 , 



2 nd . as condition of reciprocity (Ball) of a dynam X and a double 

 rotation 5, 



we make a connection between the point- and ,Sp 8 -geometry in 

 Sp s on one hand and the geometry of dynams and double rotations 

 on the other hand. 



To each theorem of the former corresponds a theorem of the 

 latter geometry. Nov the remarkable fact makes its appearance that 

 the fundamental theorems of the geometry of Sp t correspond to the 

 fundamental theorems of the theory if screws of Ball in Sp t . 



With this as basis we shall show, though it be but by means of 

 some few examples of a fundamental nature, that the principles of 

 a generalisation of the theory of screws are very easy to be arrived 

 at by transcription of the simplest properties of the point- and 

 &p H -gcometvy in >Sp 9 which examples can at the same time be of 

 service to explain the above observations on the theory of Ball in Sp z . 



To avoid prolixity we introduce the following notation. We call: 



dynamoid the system of lines whose conjugate pairs can serve 

 as bearers of a dynam. 



rotoid the system of planes whose conjugate pairs can serve as 

 bearers of a double rotation: So dynamoid and rotoid correspond 

 to dynam and double rotation as in the notation of Bali- "screw" 

 to dynam and helicoidal movement. 



Let the following transcriptions be sufficient to explain the appli- 

 cation of the above principle. 



aX: Point X bearing a mass Dynamoid X bearing a dynam 



o. of intensity X. 



aS: Sp s S with a density of Rotoid S bearing a double 



mass o. rotation of intensity o. 



(X'X")\ Right line, locus of the Pencil of dynamoids, locus of 



centres of gravity of the bearers of the resultants of 



variable masses in the two variable dynams on the dyna- 



points X' and A'". moids X' and X". 



(3'E"): ,S/vpencil. Pencil of Rotoids. 



A right line has always A pencil of dynamoids always 



