of the Second Order^ bounded ly parallel Planes. 475 



V. When the surface is an elliptic paraboloid. 



The general expression for a perpendicular from the centre 

 of a surface of the second order on a tangent plane is 



p- = a- cos^ X + i^ cos yU- + c^ cos^ V. 



Let I and I' denote the parameters of the principal sec- 

 tions in the planes of ab and a c, then b^ = al, c^ = a Z', 

 hence 



-^-3-=: cos-* A -I cos-* U.4 COS^ V. 



a'- a a 



Now when the surface becomes a paraboloid a becomes in- 



i 'i 



finite, and — , — ■ each equal to zero. Hence p, while it has 

 a a ^ 



a finite ratio with respect to c, is infinite corapared with b and 

 c\ therefore the coefiicient 



abc a/77' r. V- 



— ^ = ^ = 0, when 0=00. 



^ a cos^ X 

 Hence V = i/(A + B) (12.) 



VI. When the surface is a cone. 



We may consider the cone as the limiting surface of an 

 hyperboloid of two sheets. 



Let c be the real axe, then in the hyperboloid 

 p^ = c^cos^ v—a^ cos^ K—b^ cos^ fju, 

 or dividing by c% 



■—■ = COS^ V ~ ^°^^ ^'" ~2" *^°^^ /*» 



but when the surface becomes a cone, 



— = tan a, — = tan /3, a and /3 being the semiangles 

 of the cone, and cos^v = 1 — cos^x— cos^ /x; hence 



tan a tan /3 « /i q \ 



V= ^<(A + B)+ {i_sec2^cos^x-sec2^cosV}^ '^ ^^ 



It is plain that the difference between any slice, and the sum 

 of the cylinders on the bases of this slice, is independent of 

 the distance of the bounding planes from the centre of the 

 surface. 



VII. A plane cuts off" from a surface of the second order 

 a segment of constant volume, to find the surface enveloped 

 by tills plane. 



Let the surface be an ellipsoid, and let the volume be the 

 wth part of the semieliipsoid, then the volume of the slice be- 

 tween this variable plane and the parallel diametral plane will 



