942 



Of the 28 nodes of the envelope [L), 4 are lying at infinity, 

 8 on the circumference of the circle x^ -\- y"" =i\, 8 on the hy-perbola 



x^ — y"^ = and 8 on the hyperbola xy = — 2/. 



The 4 pairs of parallel asymptotes of {L), which correspond with 

 the bitangents of {L') passing through 0, touch the conic {K'), which 

 is the reciprocal polar curve of {K). 



The nodes of (L) lying on {C), if they are real, are for the dynam- 

 ical problem under discussion the vertices of the quadrangular 

 figures, which as appeared before, may serve as envelopes ; the 

 branches intersecting in those points meet perpendicularly, as was 

 proved for a more general case ^). 



§ 4. Asymptotes of {L). Besides the 4 pairs of parallel asymp- 

 totes, {L) has moreover generally speaking 4 asymptotes passing 

 through 0, which are perpendicular to the asymptotes of {L'). 



Of {L') two asymptotical directions may coincide. 



In this case the corresponding asymptotes do not pass through 0, 

 but they are removed from at equal distances. In that case on 

 the straight lines passing through {L) has two cusps in which 

 the straight line is a tangent. The said straight line is to be consid- 

 ered to belong to (L) ; consequently (L) is degenerate. 



Various shapes of (//). 



§ 5. The equation of {L') reads (§ 1) : 

 \qx- + 2plxy — (2p -f- q)y' + 2p|^ = s \x^ - 2(2p + 1)■^•^V'^ + y'V 



where 



s = q' —p[4.r + P). 



Its shape will in the first place be dependent on the nature of 

 the bitangents drawn from 0, viz. whether they are imaginary 

 ip <^ 0), or real {p ]> 0) and touch the curve in real points or are 

 isolated. 



Further on the nature of the conic {K) which may be an ellipsis, 

 an hyperbola or a degeneration. 



Finally on the reality of the asymptotes. 



We can prove now, that {L') has as many real asymptotical direc- 

 tions as it has pairs of real points of intersection iinth {C). 



Let {cos a, sin «) be the point of {€) lying on {L'), then we have : 



1) Phil. Mag. 1. c, p. 297. 



