OP ARTS AND SCIENCES. 323 



6°. The direction of the normal to the surface is given by ^q'iQ'^, 

 and the projection of the radius vector on the normal is 



and this is the perpendicular from the origin on the tangent plane to 

 the surface at the extremity of q. Its length is 



Tr=± ^^P'^^X (10) 



Hence, the element of volume swept by the radius vector is 





and the finite volume is 



V= i,JJS(>n\n',. (11) 



Here, again, as in 2°, a change of origin will affect the result, and 

 give a different portion of the volume. As before, suppose d to be the 

 vector of the new origin. Then the new vector, whose extremity 

 generates the surface, is w = (; — 8, and the volume swept by w is 



7". The equation of the ellipse may be written 



Q = a cos X -\- ^ sin x, (13) 



where a and (3 are the principal semi-diameters, and x is the eccentric 

 angle. T« = a, T^ = b, and a is perpendicular to /5. By differen- 

 tiation 



DjP =. q' =z — a s'm X -\- ^ cos x, 



and taking the tensor 



Tq' = sjcfi sin'^ X -j- b"^ cos'-^ x 



= sj{d^ — 6-) siu^ X 4- b\ 

 Hence the arc of the ellipse is 



X X 



So ={ T(y = \ sf{a' — ¥) sin-^ x -[- b\ (14) 



