418 B17" _ B"_y' -yraq _‘ 71:“: aIBr] _ anfla B117 _ B71: 7:1“ _ ya" auB _ aBu B71 _ ya; __ 7:“ aBr _ a:B | 36. It is important to express the nine coeflieients in terms of three independent quantities. A solution which, although unsymmetrical, is very convenient in Astronomy and Dynamics is to use for the purpose the three angles 0, 4:, -r of fig. 19 ; say 6 = longitude of the node ; <1; = in- clination —' and 7 = longitude of 2', from node. X A1 Fig. 19. The diagram of transformation then is
- c y :
'-'1 cos’? cos 9 - sin'rsin 9cos¢ cos'r sin9+ sin 7 cos 6‘ cos ¢ sin 7 sin ti) 3;: - sin ‘r cos 9 - eos'rsin9e0s¢ - sin‘rsin9 + cos 7 cos 9 cost? cos '1' sin <1’) 21 sin9 sin¢ — c0s0sin¢1 cos¢>' But a more elegant solution (due to Rodrigucs) is that contained in the diagram i 23 3/ 2 $1 1+N3-#9-vi 9-W»-V) 2(M +11)
- 2(Ap.+v) 1 -2.‘-’+p.‘-’—u‘3 2(;u/-A)
- l 2(vA 2(;w + A) 1 — A9 — p? + u?
-2-(1 +A9— p.2+v’) The nine coefficients of transformation are the nine functions of the diagram, each divided 1) 1+A2+p"’+1/3; the expressions contain as they should do the t ree arbitrary quantities A, p, v; and the identity $12 + y,9+:12=.'c9+3/“’+:.‘-’ can be at once vcrilied. It may be added that the transformation can be expressed in the quaternion form i.c, +jy1 + 1.2:, = (1 + A)(i.'c+j_r/+ Icz)(1 + A)‘1 where A denotes the vector z'A +j;u+lcv. Quaclric Surfaces (Pa7'ab0l0z'cZs, Ellz'ps0z'd, [I3/pcrboloids). 37. It appears by a discussion of the general equation of the second order (:1, . . §§:c, 3/, 2', 1)9= O that the proper quadric surfaces‘ represented by such an equation are the following five surfaces (a a11d 12 positive) :— '1. =2”. -‘i 111-‘ -. '. _ ) z 2a+2b, e ptic panbolonl (2.) 2 = ii hyperbolic paraboloid. ” go" 2b’ (3.) g+ = 1, ellipsoid. (4.) + 3% -35: = 1, hyperboloid of one sheet. (5.) :—:+ 35- - = — 1, hyperboloid of two sheets. ‘ The improper quadric surfaces represented by the general equation of the second order are (1) the pair of planes or plane-pair, including as a special case the twice repeated plane, and (2) the cone, including as a special case the cylinder. There is but one form of cone ; but the cylinder may be parabolic, elliptic, or hyperbolic. GEOMETRY [some .'.I.YTICAL. It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces. In the elliptic paraholoid (fig. 4': 20), the sections by the planes of ::.L' and :3; are the parabolas e /Z 2 ID .7.’ “=23”: as _ 4,, hav1n_9; the common axes 0:; and the section by any plane
- =-y pa)-{Vllel to that of .1-_,) is the / 0
ellipse so that the surface is generated Hg‘ 20' by a variable ellipse moving parallel to itself along the parabola: as direetrices. Z In the hyperbolic paraboloid (fig. 21) the sections by the planes of
- c, zy are the parabolas
— ::= -9“? hav- 2a’ 2b’ ing the opposite axes Oz, 0:’, and the section by a plane z= 7 parallel to that of my is the <63 _ 2/'3 2b’ whichhas its transverse axis parallel to 0.7; or 03/ according as -y is positive or negative. 2’ The surface is thus Fin. 01 generated by a variable °' " ' hyperbola moving parallel to itself along the parabolas as direc- 3/ .1‘ E ‘lb. 31 Ac...
- 4
hyperbola 7 = ‘~~.-_-- trices. The form is best seen from fig. 22, which 1'cp1'Os(-ms the sections by planes parallel to the plane of a'_I/, or say the contour lines; the _ ‘ continuous lines are “’ the sections above the plane of 9:3/, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle. In the ellipsoid (fig. 23) the sections by the planes of ca-, :1 , and a:_2/ are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be con- sidered as generated Fig. 23. by an ellipse moving parallel to itself along two ellipses as dircctrices. In the hypcrboloid of one sheet (fig. 24), the sections by the planes of 1:12, :3; are the hyperbolas
having a common conjugate axis :02’; the section by the plane of