( 823 ) 



So if there arc among n lines ;it most JO belonging to a linear 

 complex we can satisfy the equations (9) by choosing all intensities 

 except those belonging - to these 10 equal to and then (if not all 

 subdeterminants of order 9 tend to Ü) we shall be able to bring along 

 these last only one distribution of intensity differing from in such 

 a way that the system of forces obtained in this manner is in 

 equilibrium. 



We have thus at the same time arrived at the following theorems : 



For the equilibrium of ten forces it is necessary (hot these belong 

 to one and at most to one linear complex. In this case always one 

 and not more than one distribution of intensity in possible. 



If we continue the investigation of the equations (9) we then 

 obtain successively the conditions of equilibrium of 9, 8, 7, 6, 5 

 forces. We can express the result as follows : 



In order to let n forces, 11 > n ^> 4, admit only of one distri- 

 bution of intensity in equilibrium, it is necessary and sufficient for 

 them to be the common elements of exactly 21 — n linear complexes. 



In particular for n = 5 we find the condition that the forces must 

 belong to a system of associated lines of Segre. 



This has given us a connection with a former paper in which we 

 treated .this case synthetically. 



V. The condition that ten forces in equilibrium belong to one 

 complex follows almost immediately out of the interpretation of the 

 equation .£'«ypy = as condition of reciprocity of force and double 

 rotation . 



Let e.g. ten forces be given in equilibrium ; nine of these forces 

 chosen arbitrarily determine a complex, so also the double rotation « 

 for which none of them can perform labour. The united system of ten 

 forces, as being in equilibrium doing no labour for no motion 

 whatever, it is necessary for the tenth force to be likewise reciprocal 

 with respect to the double rotation a, i.e. this force belongs with the 

 former nine to the selfsame complex. 



Equally simple is the deduction of the conditions of equilibrium 

 for nine forces. 



For eight forces determine a simply-infinite pencil of complexes 

 whose conjugate double rotations « -f- Id are all reciprocal with 

 respect to these eight forces. So they must also be reciprocal with 

 respect to the ninth force in equilibrium with these, i. o. w. the 

 latter must belong to all linear complexes to which the eight others 

 belong. 



And so on. 



