OF ARTS AND SCIENCES. 339 



V = f //Shx ^^"^ [Cha;] '. (53) 



y X 



Similarly the equation of the imparted hyperboloid gives 

 ^qq\q\ = — ChxSo'ciG = ChxSaVa'ff 

 = ChxSay^ = — abcChx, 



and 



r=^JJchx= $^^[Shx]^ (54) 



28*^. Formulae (52), (53), and (54), it will be observed, give the 

 volume of the portion of space swept by the radius vector ; i.e., of the 

 space contained between the surface of the quadric and the two cones 

 whose vertices are at the centre, and which are generated by the limit- 

 ing positions of the radius vector at the limits of integration ; or, if 

 the lower limit is zero, the volume contained between the surface and 

 a single central cone. The volume of the segment cut off by a plane 

 perpendicular to an axis, can be found by finding the difference be- 

 tween the volume given by the general formula and the whole volume 

 of this cone considered as limited by the plane ; but more easily and 

 simply by a change of origin, according to the method of 6^', by the 

 use of formula (12). 



29''. Beginning with the ellipsoid, let the origin be transferred 

 to a fixed jjoint on a, and let w be the new generating vector. Then 



v7 =1 Q — ma, 



m being a scalar. Formula (12) becomes in this case 



y X X y 



We have, as in 24*^, 



q\^= — « sin a; -|- (7 cos x, 



q\ = a' sin x, 

 hence 



mSaq'-^o'^ = »jS.«( — a sin a: -[- cr cos x)a' sin x 



= m sin X cos xSa^y = mahc sin x cos x. 

 In 26°, we found 



^QQ'iQ'^ ^^^ "^^ ^"^ ^' 



