OF ARTS AND SCIENCES. 329 



If the sector be cut with one boundary parallel to a, so that at the 

 upper limit x = m, while x^ is taken = 0, theu 



^ =: — - sm '^ — -z~ = — - sin '^ . (3o) 



2 a|_2 6J 6 « ^ ' 



This is one-third of the parallelogram formed by the co-ordinates of the 

 point at the extremity of q. 



15°. The equation of the cycloid gives the following : — 

 Q = a(d — sin 0) + ^(1 — cos d), 

 q' = «(1 — cos 0) -\-§ sin d, 

 TVqq' = TV[a^{9 sin G — sin^ ^ — {1 _ cos 5 }2)] 

 = 0^(0 sin d — i sin2 i d). 



Hence 







(d sin ^ + 2 cos 6 — 2) 



= Y [sin ^ — ^ cos (9 -f 2 sin ^ — 2^1 



= Y[^^'^^d — dcosd — 2dy. (36) 



The complete area is ^ = SOa'^. 



IG'*. In 14°, we found for the parabola 



? = Y " + ^^' 



T(>' = \/d'x^ -f- b\ 

 Revolve the curve about ^, the tangent at the vertex. Make 



u = — . 

 2 



The area of the surface thus -generated will be [see (5)] 



X X 



S=(Ii j uTq' = di ~ ja; V^V+T^ 



Xo Xq 



^ fj^ r {2a'^x^-^h^').rsjn2x^-^b^ _ J^ gj^_i ^T 

 L Ua 16a2 bj 



