of the Second Order, bounded hy ■parallel Planes. 473 



I. Let ab c he the semiaxes of a surface of the second 

 order, a' U the semiaxes of a diametral section of the surface 

 parallel to the given planes, d the semiconjugate diameter to 

 this section, p the perpendicular from the centre on the tan- 

 gent plane parallel to these planes, a and /3 the semiaxes of 

 the section of the surface made by any parallel plane ; let u 

 and -sx denote the segments of c' and p between the centre of 

 the surface and this plane, putting m" and iJ for the segments 

 of d between the centre and the given bounding planes, and 

 calling the volume of the slice V, we shall have 



V = 7r /rf^[«/3] (1.) 



k/ -us' 



or changing the independent variable, 



= TT I dTsl^-l 

 k/ -us' 



ig the indepenc 



Let a plane section of the surface be drawn through a' and 

 c', then 



oL'.dw V d~—u^ : d ; 

 similarly, 



fi:V:: -/ d^-u^: d. 

 Multiplying these proportions, 



«^= 4^(^''-«') (3.) 



Now M : cr : : dip; hence —5 — = ^, and a' 6' » = abc, 

 ^ du d 



by a well-known property of surfaces of the second order. 



Making these substitutions in (2.). 



V = ^3' ^ fdu [d'-ti^]; (4.) 



•y u' 

 or integrating and taking the limits, 



V = -^n^d-{u"-n') - ^ 3}' • • • • (5-) 



or V = ^f- 7: {u" - 21') /e c'2 - 2 m"2 + 2 u" u!-2u^\. 



Adding and subtracting u"^^ + ti'^^, this equation may be trans- 

 formed into 



V = — |7r(z/'-M'){3(c'2-?i"2) + 3(c'2-«'2) + (M"_M')2|. (6.) 



Let «" /3", a! j3' be the semiaxes of the sections of the surface 



