326 PROCEEDINGS OF THE AMERICAN ACADEMY 



ll*'. The scalar equations of tlie cycloid are 

 X = a (d — sin 0), 

 y =z a {\ — cos 0). 



Let T« = T|3 = a. Then the vector equation of the cycloid may be 



written 



Q = xU« + yUt3 = a{d — sin d) + |3(1 — cos 0). (24) 

 "Whence 



q' := «(1 — COS 6) -j- ^ sin 0, 



(/2 = «2(l _ COS ey -f ^'' sin^ 0, 

 TiJ = a\Ji — 2 COS ^ -f sin- 6 + cos- 



= ay/2 — 2 cos d = 2a sin -|^. 



^ e 



... s — s„ = 2« / sin ^5 = 4a [cos ^oj. (25) 



For the complete arc 



s — 5p = 4a COS Id ,^ „ = 8a. 



12°. The areas of the conic sections are easily obtained from for- 

 mulas ('3) and (4). For the ellipse we have, a and ^ being any two 

 conjugate semi-diameters, 



Q =z a cos a: -|- (3 sin x, 



q' = — a sin a; -^ j3 cos x, 

 whence 



TYcq' = TV(a|3 cos- x — (-ia sin^ x) = TV«|3 ; 



Or, if « ± /5, 



Xo 



= ^sml[x-x,'\. (20) 



The whole area is Qab. 



We may change the origin to any fixed point, and so obtain the arga 

 of any portion of the ellipse. Suppose e is the vector of the fixed 

 point, and nr the new radius vector from this point; then the equation 

 of the ellipse becomes 



w = (p — e) = a cos x -\- ^ s'w x — s. 



