476 Volume of a Segment of a Surface of the Second Order. 



be ——-abc (1 —n), - ^ aba being the volume of the ellip- 

 soid ; to determine the volume of this slice ; in equation (5.) 

 let u' = 0, and we shall have 



V 7 f "" "'" T 



u" 



let —r- = 4' , then 

 c 



y = abc 



'^ ("- 4)- 



2 w 

 But V is also = — — a b c(l — ?j). 



Equating these values of V, 



x^—3x + 2(l-w) = (14.) 



Now as this cubic equation falls under the irreducible casCy 

 assume the formula 



. o 3 . sin 3 4) 

 sm'^ (f> 7" ^^" ^ "I aT^ ~ ^> 



or, multiplying this formula by 8, 



(2 sin cf))3— 3 (2 sin 4)) + 2 sin 3 <J. = 0. 

 Comparing this equation with 



aP—3 X + 2 {1—n) = 0, 

 we shall have a; = 2 sin 4>, sin 3 <fi = 1—n; 



U" CT 



but X = 2 sm <B = -7— = — , 



^ c p 



hence x^ = 2 p sin <p. 



In the ellipsoid, 



p^ = a^ cos^ X + b"^ cos^ fJ' + c~ cos- v ; 

 or multiplying by 4 sin- ^, and putting ot for 2 p sin 4> 

 ra-^ = [2 a sin 45)^ cos^ X + (2 i sin <p)- cos- /i, + (2 c sin (f )- cos^ v, 

 therefore ct the perpendicular on the enveloping plane, is a 

 perpendicular on the tangent plane which envelopes the ellip- 

 soid whose equation is 



x^ v^ z^ 



a^ W c^ 

 the equation of an ellipsoid similar to, and similarly situated 

 with the given one. 



Let n = i, then sin 3 <$> = (1 —n) = i •/ 3 cfi = 30°, or 

 <}> = 10°, and the locus in this case is 



4 + 7I- + 4 = * sin^ 10°. 

 c^ b^ c" 



