( 215 ) 



(U TP y dif dy ,,r .,. , 



Now dE is equal (o — (1), so — — ziz -i— --^ ; 1/ -—=/'/, it we i»io(luee 

 1/ 1 A C/A (Lv. dl' 



QV l tangent SP as fai- as the intersection f willi ]*I // j:-a\\^. So 



TP _PI 



'TN~~QN^' 



That is: the poinl of contact T in (inc^lion is the point where 



/*^Y is intersected by QV- 



10. Ellipse on one 0/ its a.res as recti/i/iii</ curre. If \vq take 



for the constant [)oint X of tlie .*-a.\is (9) tlie centre O of 



the ellipse (tig-. 7), then the displacements of (-^ and /^ are respect i\ely 



- d.c . — .V d.v . 

 Q 0. (h = ,r (h = .V and di/ ; llie (piolient — is acconhng to 



// ' y (ly 



a' 

 the central en nat ion of the ellipse constant := — . So the point ot 



b 



contact of the right line OP remains dnring the whole mofo:i a 

 tixed point; this point of contact is the point R where OF is inter- 

 sected by QV (9); so this point of intersection remains a fixed point 

 dnring the whole of the motion. The qnotient of the displacements 

 of and F is RO: RF, so RO: RF=cf:b\ 



In order to find the natnre of the rectified curve, making use of 

 this tixed point R, we determine ^) of FQ, considered as similar 

 sj'Stem, the points moving perpendicularly on their radius vector 

 out of R; these points T prove to be real for the ellipse and lie in 



FT ^> , . . 



such a wav that — - = ± — ; their distance to A remains constant 

 TQ a 



daring the whole of the motion ; they describe a circle \vitli centre R. 



If we produce RT till it intersects the .I'-axis in ^"^ and the //-axis in Y, 



then TU and UY also remain constant during the entire motion. 



From all this ensues that the rectified curve is an epi- or hi/poci/cloid 



with R as centre; 7' describes the circle of the basis. 



11. To find the length of the radii yt7'and h 7'6^(fig. 7) expressed 



in the half axes (t. and of the ellipse, wc presuppose the figure in 



such a position where R has arrived in the production of the small axis ; 



RF b' FT b 

 we make use here of the abo\e utixen relations — = — and — : = — . 



" RO a^ TQ a 



Without any difücnlly we liiul for the radius of the rolling circle 



ab 



=zrT u = 



2 {a + b) 



1) We give here for sliorlness'sako the resulls only. 



