178 LiPie\. 



I,." 2 



(33) t"'= A + Bx"^ Cv 



Theorem 4 is applicable here, hence we have 



Theorem 16. The " n " system possesses Projjerty I. 

 For the " n " svstem, 



(83) C = ^ 



xp - (j>v' 





\ 



hence the bisection property II does not hold unless n = 2, i.e., for 

 dynamical trajectories. For the general " n " system the bicircular 

 quartic of Property I has a node at the given point, one of the tangents 

 having the direction of the initial element v', and the other having 

 the direction. 



^ ^ ^ C'(l+/^)-6.' 



where C is given by (83). Substituting this value and introducing 

 the direction of the force vector ^/0 = co, we may WTite (84) as 



(85) f^ =.!i±i i^, 



1 + coi' 3 1 + v^ 



or 



(86) '""^^ " + ^ 



tan Bo 



where di is the angle between the initial direction and the force vector, 

 and 02 is the angle betw-een the direction of the second tangent and the 

 initial direction. Conversely, we may easily show that if the angles 

 6 1 and do are related as in (86), C must have the value given in (83). 

 We thus have 



Theorem 17. Property ir. For an "n" system, the focal locus 

 described by Property I has ttco distinct tangent lines at the initial point. 

 The tangent of the angle which the initial elevwnt makes ttiih the force 

 vector is to the tangent of the angle tchich the second tangent line makes 

 with the initial element as n -\- 1 is to 3. 



We may easily show that the most general triply infinite system of 

 curves possessing Properties I and IP has an equation of the form 



v'" = A + Bv" + — ^ (2 - n) f:p^ - 3 i'"^ 



a; - / ( "^ 1 + v' 2 ^ 



where a fixed direction co (u, v) replaces the direction of the force vector. 



