939 



In wliat tbllovvs for S=2, the ease of the equality of frequencies, 

 the envelope will be treated anew and from a more geometrical 

 point of view. 



We shall therefore have to occupy ourselves with the envelope {L) 

 of the system of ellipses 



•'• = l/? cos t , // = l/ïZr^ cos {t—(fi), 



in which $ and if are two variable (pumtities in the most general 

 case connected by the relation : 



(1 - C) cos <i =~ ^ / ± \/ p S-' + <fi I r -V- - l\ 



By elimination of t we tiiid for the ecjuation of the ellipses 

 {A) (1 - ?) x'-^l \/l{\:^cos if . .vy + ?.;/•-■ = ; (1— g) sm^ r/. 



Let us now determine the reciprocal polar curves of these ellipses 

 with respect to the circle : 

 (C) .v^-^f = h 



in the circumference of which circle the vertices are situated of the 

 rectangles, which are circumscribed to the ellipses (^4), and have 

 their sides parallel with the axes. The envelope {L') of this new 

 system of ellipses will be the reciprocal polai- curve of the envelo[)e 

 (L) wanted. (Cf. note p. 943). 



The new system of ellipses appears to be given by : 



(.4') W' + 2 y/gj^) cos -p . ,vy + (l-?).v'^ = 1. , 



By elimination of <f between this equation and the given relation 

 between ti and ^ we tind : 



^(p^' + q^ -I- r 4- i r^ ^ ,ry-^ :.= |1 - W + /.ft/ - (1 -?) t/T- 



This equation contains ? to a no higher order than two. The 

 equation of the envelope of (.4') may consequently be written down 

 at once. After some reduction this equation of (/>') becomes : 

 -(4r ^l''){y''—xy — 4.p(\^lxy—y')' j 4.p{Ar + l'')x-y-= 



=\q\vY' - 4^(1 ^-l^vy-y^){y^ -x'). 



As [L') is now apparently of the fourth order, the envelope [L) 

 wanted is of the fourth class. 



As {L') in general as we shall see, has no double points or cusps, 

 it has been determined by this, that (/>) is of the twelfth order. 



{L), like (/>'), has the origin as centre. 



If we multiply the equation found for [L') by p, (the cases /? = 

 and /> =rr X we shall consider sepaiatcly in § 9) it appears that it 

 may be written thus : 



61 



Proceedings Royal Acad. Amsterdam. Vol. XVI. 



