474 Prof. Booth on the Volume of a Segment of a Surface 



made by the bounding planes, and t the distance between 

 them, or the thickness of the slice, then 



a be a be 



and {u"—u') :t::c':p. 

 Making these substitutions in (6.), we shall have 



or putting A and B for the areas of these sections and S for 

 the sphere whose diameter is t, 



V = i^(A+B)+ (^')S (7.) 



or the volume of any segment of a surface of the second order 



bounded by parallel planes is equal to one half the volume of 



the tnoo cylinders whose bases are the sections of the surface 



made by the bounding planes, and common altitude the thickness 



of the slice, together mth the sphere whose diameter is this thick- 



n. , ab c 

 ness multiplied bj/ the constant coefficient — 3—. 



Jr 



II. When the surface is a sphere, ass b = c —p, and 



V = i^(A + B) + S, (8.) 



the formula given by Legendre for the volume of a slice of a 

 sphere. 



III. Wheri the surface is a discontimious hyperboloid or one 

 of two sheets. 



The semiaxes in this case are a V — 1, b */ —\ and c, the 

 formula (7.) therefore is changed into 



V = i^(A + B)-(^')s (9.) 



Should the planes cut the surface in hyperbolic sections, p 

 becomes imaginary, and the expression in this case fails as it 

 evidently should, A and B, as also V, becoming infinite. 



IV. When the surface is a continuous hyperboloid or of one 

 sheet. 



The semiaxes are a, b, c v^ — 1> and the formula becomes 



V = i^A + B) + ^—1 (^^) S, (10.) 



an imaginary expression; but when p is imaginary, the ex- 

 pression for V is real, as it evidently should, the sections A 



and B being in this case real, and therefore V. 



For putting p \^ —l for p, p^ is changed into — p^ V' — 1> 



and V = iMA + B)-(^')s (11.) 



