SOLID ANALYTICAL] into the foregoing form, and hence that the two linear equations represent a line. Secondly, taking for greater simplicity the pointQ to be coin- eideiit with the origin, and a’, 3’, 7', p to be constant, then 19 is the perpendicular distance of a plane from the origin, and a’, 1-3’, 7’ are the cosine-inclinations of this distance to the axes (a""+3h‘,+'y'3= 1). I’ is any point in this plane, and taking its co- ordiiiates to be (.r, 3/, 2) then (g, 7;, g‘) are=(.1:, y, .7), and the fore- going equation p=a’§ + 3'71 +7’; becomes a'-'3 + 3'3! + '/'7 =2’ ; which is the equation of the plane in question. It", more generally, Q is not coincident with the origin, then, l:ll'lllg' its coordinates to be (c(., I), c), and writing pl instead of 1), the equation is a’(.ir: — a) + fl’(y — 12) + 7/(2 — c) =pl ; and we thence have pl =32 —-(rca’+bfl'+c-y’), which is an expression for the perpendicular distance of the point (a, I), c) from the plane in question. lt is obvious that any linear equation A.L‘+ B3/+ Cz+ D=0 between the coordinates can always be brought into the foregoing forni, and hence that such equation represents a plane. 'l'hirdl_v, supposing Q to be a fixed point, coordinates (It, I), c) and the distance Ql’,= p, to be constant, say this is=d, then, as ln_-fore, the values of 5, 7;, {are a:— rt, 31- I2, z—c, and the equation 5’ + 77" + {'3 =p'~’ becomes («"1 - 0)” + (1! - 5)’ + (2 - 0)” =d2. which is the equation of the sphere, coordinates of the centre = (a, I), c) and radius=tl. A quadrie equation wherein the terms of the second order are .:;'-’+ _z/'-’+;"-’, viz., an equation
- 2.-2+y'3+z'3+A.r+ B1/+Cz+I)=0,
can always, it is clear, be brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre — Jlul, — .';B, -— .130-, and squared radius = ;';(A” + B2 + C3) — D. Czlimlcrs Cones Ruled Sm‘ aces. 7 7 A singly infinite system of lines or system of lines depending upon one variable parameter forms a surface; and the equation of the surface is obtained by eliminating the parameter between the two equations of the line. If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a given line, then the surface is a cylinder. Begiiiiiing with this last case, suppose the lines are parallel to the line a'=-mz, 3/=1z.:, the equations of a line of the system are a:=m:+_a, 3/=n::+b,—wlierc (1., b are supposed to be functions of the variable paranieter, or, what is the same thing, there is be- tween them a relation fie, _b)=_0: we have a=a:—mz, b=y-— ng, and the result of the elimination of the parameter therefore is _f(.:r— 121:, ,1/— 71:)=0, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line .::=m:, 3/=-nz. The equation of the section by the plane :=0 is jgfrr, y)=0, and conversely if the cylinder be determined by means 0 its curve of intersection with the plane z=0, then, taking the equation of this curve to be flag 3/)= 0, the equation of the cylinder ]_' _/"(.7--— 772:, 3/~-nz)=0. Thus, if the curve of intersection be the circle (.r: — a)9_+ (y — 1-3)‘-’=-y9_,we have (.1: — -mz —_a)2 + (y — 722- 1-3)”:-79 as the’ equatioir of ‘an oblique cylinder on this base, and thus also (.r — a)‘-+ (_r_/ — 3)‘-=-r as the equation of the right cylinder. lf the lines all pass through a given point ((1., b, c), then the equations of a line are a:—a=a(z—c), 3/— b=[-3(z— c), where a, [-3 are functions of the variable parameter, or, what is the sai_ne_thii_ig, there exists between them an equation f(a, fl)=0; the elimination .v-— a 3/ - b _ , __
- —c z—c
equation, or, what is the same thing, any homogeneous equation _§’_.v— cc, _7/— b, z—c)=0, or, taking f to be a rational and integral niietion of the order 7L say (*)(.e — a y — b 2 —c)"=0 is the general equation of the cone having the point (a-, 12,, c) for its vi-rte.’. Taking the vertex to hp at the origin, the equation is (*)(a*., 3/, :)"=0; and, in particular, ( _)(;e, 3/, :)”=0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex. of the parameter gives, therefore, f ( =0; and this 34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or 7'.r2_r/uhcs). If the system be such that a line does not intersect the coiisecu— tive line, then the surface is a skew surface, or scroll ,' but if it be such that each line intersects the consecutive line, then it is a developablc, or torse. GEOMETRY 4 1 7 Suppose, for instance, that the equations of a line (depending on - . 9‘ 2’/_ 3/ 23 Z 1 the variable parameter 0) are E + ;—0(1 +5 , 3- E =§(1 —g) , . . . ac’ :2 2 2:’ 3/2 22 then, ehiinnating 0, we have a§— 5-2=1 — is , or say 33+ be -— c—,=1 , the equation of a quadrie surface, afterwards called the hyperboloid of one sheet ; this surface is consequently a scroll. It is to be re- marked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (de- pending on the variable paraineter ¢>) are 7‘ Z _ 2 E E -1 "J -¢(1-1))» an-¢(1+z)~ It is easily shown that any line of the one system intersects every line of the other system. Considering any curve (of double curvature) whatever, the tan- gent liiies of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the systeni,—tliat is, they form a developablc, or torse; the curve and torse are thus iii- separably connected together, forming a single geometrical figure. A plane through three consecutive points of the curve (or oseu- latiiig plane of the curve) contains two consecutive tangents, that is, two consecutive lines of the torse, and is thus a tangent plane of the torse along a generating line. Tra22.,sy"0rmat:'0n of Cloorclinates. 35. There is no difficulty in changing the origin, and it is for brevity assumed that the origin remains unaltered. lVe have, then, two sets of rectangular axes, 0.7:, 01/, Oz, and Owl, Oyl, Ozl, the mutual cosine-inclinations being shown by the diagrain-— if as v : I a 3 '7 a’ 3' l 7' ' a" | 3" . '7” l -lll 3/1 ~ "1 that is, a, ,8, 7 are the cosine-inclinations of Oxl to 09:, 03/, Oz ; a’, ,8’, 7’ those of Oyl, &c. And this diagram gives also the linear expressions of the coordinates (xl, 3/l, zl) or (.7/;, y, 2) of either set in terms of those of the other set ; we thus have
- = aa:l + a’yl + a"zl ,
y=B9:l+fl’,3/l+fl,’:zl ; ‘=7-151+?!/1+7 31: which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the before-mentioned equation 1) = a'£+ ,8’17+'y'§, adapted to the problem in hand. But we have to consider the relations between the nine coeflicients. By what precedes, or by the consideration that we must have identically x2 + 3/2 + 22 = xl2 + 3/l2 + zl‘2, it appears that these satisfy the relations— 9.’l=a :z'+B 3/+7 2, yi=a’w+B’y+~/2, zl=a”ac+/3 y+-y”z, a2 +3” +7” =1. a”+a'2 +a"’ =1.
- 1’? +3’? +7” = . a'3+B"= +3"? =1.
any +3112 +7172 ___ , 7:7.+.y’2 +,y"2 = 1 ’ a'a"+3’a"+7’7”=0, Bv+Bj7,'+afj-y,j’=9, a"a+B”B +7”'r =0, 'ra+'r,a,+'r,,a,,=U. aa’ +38’ +77’ =0, al‘3+al'3+al'3 = , either set of six equations beiiigimplied in the other set. It follows that the square of the determinant a I, 3 I, 7' all’ fill’ ‘Y’! a 2 3 7 7 is=1 - and hence that the determinant itself is =;{;1. The dis- ’ a - 0 I 9 tinction of the two cases is an important one: if the determinant is = + 1, then the axes Owl, O3/l, O:l are such that they can by a. rotation about 0 be brought to coincide with 03:, O2, 2 respect- ively; if it is = — 1, then they cannot. But in the latter case, by measuring a:l,yl, zl in the opposite directions we change the signsof all the coefficients and so make the determinant to be = + 1 ; hence this case need alone be considered, and it is accordingly assumed that the determinaiit is = + 1. This being so, it is found that we Iiave a further set of nine equations, a= 1-3'7 ’ — 1-3"-y’, &e.; that is, the coeffieicnts arranged as in the diagram have the values
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