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satellite point of P; this will henceforth be indicated as the 

 satellite point *S of P. 



If q is even and consequently p odd, none of the q — 2 roots 

 of the equation (2) different from 1 is real and a point P does 

 not possess real satellite points. 



For any imaginary satellite point corresponding to a complex 

 root of equation (2) the same pi-operlies apart from reality hold 

 good as for a real satellite point. 



In the equation (2), t, a, and b occur no more, the value v, 

 therefore, is independent of t, of a and of b. For any point P on 

 each of the curves C {/>, q) the parameter of the satellite point is 

 found by multiplying the parameter t of P by the same number v, 

 which is a root of equation (2) different from 1. 



§ 3. Let / be an arbitrary straight line: 



Aa^ -f % + C = 0, 

 and Pi {.v; , yï) an arbitrary point of /. 



The satellitepoint S; of Pi has the co-ordinates : 



xivp and yivf. 

 and lies therefore on the straight line: 



A B 



~ .V -\ V + C — 0. 



This straight line /' is called the satellite of /. 



\ï Pi describes the straight line /, »S/ describes /' and conversely. 

 And as to each point Pi corresponds one point »S/ and conversely, 

 the straight lines Pi Si envelop a conic. As both the .I'-axis and the 

 y-axis, as well as the line at infinity, are positions of Pj Si, we 

 have the proposition : 



The tangents in the points Pi of an arbitrary straight line /drawn 

 to the curves C{p,q) passing through those points envelop a parabola, 

 touching the axes of co-ordinates, the straight line / and its satellite /'. 



The coefficient of direction m of / is ; the coefficient of direc- 



P 



A 

 tion ju' of r is: — — v'i—p. We have therefoi'e 

 B 



m' ==: m v9 — t'. 



Consequently there exists a constant ratio between the coefïicients 

 of direction of a straight line and its satellite. 



§ 4. Let S■^ be the intersection of / with its satellite I'; as S^ 

 is a point of /, the satellite point of /S\ lies on /', and as the satel- 



73* 



