93} 



SURFACES OF THE SECOND DEGREE. 



SURFACES OF THE SECOND DEGREE. 



P34 



to which form any equation of the second degree between three 

 variables may be reduced. These surfaces hold the same place among 

 surfaces which is held by curves of the second degree, or conic sections, 

 among curves ; and every section made by a plane with any surface of 

 the second degree must be a curve of the second degree. The following 

 article is intended entirely for reference, as the books which treat on 

 the subject hardly ever give the complete tests for the separation of 

 the different cases from each other. 



1. The preceding equation may be wholly impossible, or incapable of 

 being satisfied by any values of x, y, and ?. This happens when the left- 

 ide can be resolved into the sum of any number of squares which 

 cannot vanish simultaneously. 



2. It may represent only one single point. In this case the left-hand 

 aide can be resolved into the sum of three squares, which vanish 

 simultaneously for one set of values of x, y, and r. 



3. The equation may belong to a single straight line. In this case 

 the left-hand side can be resolved into the sum of two squares. 



4. The two last cases have a particular case which is algebraically 

 very distinguishable from the rest, though it can only be geometrically 

 represented by saying that the point or line is at an infinite distance 

 from the origin. 



5. The equation may belong to a single plane. In this case the 

 left-hand side is a perfect square. 



6. Or to a pair of planes, either parallel or intersecting. The left- 

 haml side can then be resolved into two different factors of the first 

 Stan . 



In the preceding cases there is no other surface than can be repre- 

 sented by one or several equations of the first degree. We now come 

 to the cases in which new surfaces, not plane, are generated. But we 

 may first observe that the left-hand side of the equation has a property 

 much resembling a celebrated one of integer numbers. If it be the 

 if any number of squares exceeding four, it may be reduced to 

 the sum of four squares at most. 



7. The equation may belong to a cone, having for its base any one 

 (if the conic sections. But in every case the same cone may be 

 described by a circle only : that is, every cone of the second order is a 

 circular cone, right or oblique. In this case, the first side of the 

 equation takes the form i-+<fvf, or P ! -q 2 -R 1 , P, q, and R being 

 expression* of the first degree, of the form AJ + By + cz + E. 



8. The equation may belong to a cylinder having for its base any 

 conic section. But the elliptic, parabolic, and hyperbolic cylinders are 

 perfectly distinct. In this case the first side of the equation can be 



.-.1 to the form P : + m<? + nq, P and q being expressions of the 

 firnt degree, and m and n constants. 



'.'. The equation may twlong to an elliptoid, a tingle hyperboloid, a 

 ', mi illi,,li- jHiriM'iid, or an hyperbolic parabdoid. 

 These five are the distinct surfaces of the second degree, answering to 

 the three distinct curves of the second degree namely, the ellipsoid 

 to the ellipse, the two hyperboloids to the hyperbola, and the two 

 paraboloids t<> the parabola. They will presently be further described ; 

 in the mean time the forms to which the left-hand side of the equation 

 may be reduced in these several 



parentheses means that it is a necessary consequence of what precedes 

 or is not independent. 



Ellipsoid 



Single Hyperboloid p' + q 1 H*-m*. 

 I>.nible Hy[*rbnloid p*-q* R m 1 . 



Elliptic Paraboloid p' + <j* + mn. 

 Hyperbolic Paraboloid r* q' + mn. 



The conditions under which (the several cases are produced ore 

 exhibited in the following table. Let 



) a"-r(-&' 1 ) V-KaJ 

 Vf + 2 (eV-W) <T<f + 2 (a'6'-0 a"/," 



.--*., 



When v, = 0, v, is a perfect square : if v t alao = 0, the three expression 



* - 2a7, Y' 



are all equal. Let either of them, with its sign changed, and increase 

 by /, be called w'. Again, when any three of the six quantities 



ic-o", eo-*' 1 , oi-c' 1 

 b'c'-aa', Sa'-bb 1 , a'b'-c<S 



vanish, the other three alao vanish. Let these vanish, and also let a' 

 I", f be in the proportion of a, c', I/, or of e' t b, a', or of 6', a', c. Whe 

 thin hapten, the three following 



o" V r" 

 -i t 

 ahc 



are eqnM : let either, with ite sign changed, and increased by /, b 



called w". The table is then as follows, in which j> means either o 



.ns -t- or - , and n menns the other ; and a supposition put 1 



For example, it is the condition of an ellipsoid that w and V 3 should 

 >e finite with different signs, that v, should be positive, and v t of the 

 jame sign as v, : it is the condition of intersecting planes that w should 

 iave the form 0-^-0, or that v t and V 3 should both vanish : that \V 

 hould also vanish ; and that v. should be negative. It is the condition 

 f a single hyperboloid, if v, be positive, that w and v, should both 

 ift'er in sign from v, ; but if V 2 be negative, it is enough that w and 

 IT, should have the same sign. All that precedes is equally true 

 whether the co-ordinates be oblique or rectangular ; but the following 

 is only true for rectangular co-ordinates : if the surface he a surface of 

 evolution, it is necessary that 



Vd-aa' f'a'-lll o'i'-o/ 



The forms of the ellipsoid and of the two hyperboloids may best be 

 conceived by means of the particular cases in which they are surfaces 

 of revolution. Let an ellipse revolve about one of its axes, and let all 

 ;he circular sections be flattened into ellipses : the result will be an 

 ellipsoid, derived from its particular case, the spheroid. Let an hyper- 

 bola revolve about its minor axis ; the two branches will generate only 

 one branch of a surface: let the circular sections be flattened into 

 ellipses, and the result is the single hyperboloid. Let the hyperbola 

 revolve about its major axis : the two branches will generate two 

 branches of a surface ; and if the circular sections be flattened into 

 ellipses, the result Is the double hyperboloid. For the elliptic para- 

 boloid, let a parabola revolve about its principal axis, and let the 

 circular sections become ellipses. The hyperbolic paraboloid has no 

 surface of revolution among its cases, but its form may be conceived 

 as follows : Let two parabolas have a common vertex, and let their 

 planes be at right angles to one another, being turned contrary ways. 

 Let the one parabola then move over the other, always continuing 

 parallel to its first position, and having its vertex constantly on the 

 other : its arc will then trace out an hyperbolic paraboloid. 



The ellipsoid and the two hyperboloids have centres, but neither of 

 the paraboloids has one. The surfaces which have centres possess aa 

 infinite number of triple systems of diameters having properties corre- 

 sponding to those of the conjugate diameters of an ellipse and hyper- 

 bola. These we shall not describe, but shall proceed to point out how 

 to determine the position of the centre and principal diameters or axes 

 (that is, the system of conjugate diameters, each of which is at right 

 angles to the other two) in either of the surfaces having a centre. 

 Resuming the original equation, and the co-ordinates being supposed 

 rectangular, the co-ordinates of the centre, x, T, and z, are thus deter- 

 mined. They are fractions whose denominator is v s , and whose 

 numerators are 



(bca^a" + (a'b'cc')l" + (c'a'bb')c" 

 (cab'-)b" + (b'c'-aa')c" + (a'V + cc')a" 

 (oi-c' 2 )c" + (c'a'-bb')a," + (b'c' -bb')b" ; 



and if the origin be removed to the centre, the axes retaining their 

 original directiuns, the equation of the surface becomes 



aa? + bif + c + Va'ys + 26'az + Zc'xy + w = 0, 



where w is the expression already signified by that letter, and will be 

 found to be also xa" + vft" + zc" +/. 



Let the three principal axes now make angles with the axes of .r, >/, 

 and t, as follows : The first, angles whose cosines are 0,0,7; the 



