416 Gr E O M 28. In plane geometry, reckoning the line as a curve ' of the first order, we have only the point and the curve. 111 Solid geometry, reckoning a line as a curve of the first order, and the plane as a surface of the first order, we Z 3/ Fig. 18. lia'e the poiiit-, the curve, and the surface; but the in- crease of complexity is far greater that would hence at first sight appear. In plane geometry 3. curve is considered in connexion with lines (its tangents) ; but in solid geometry the curve is considered in connexion with lines and planes (its tangents and osculating planes), and the surface also in connexion with lines and planes (its tan- ge::t lines and tangent planes) ; there are surfaces arising out of the line—cones, skew surfaces, (levelopables, doubly and triply infinite systems of lines, and whole classes of theories which have nothing analogous to them in plane geometry : it is thus a very small part indeed of the sub- ject which can be even 1'c.fcrI‘ed to in the present article. In the case of a surface we have between the coordi- nates (.r, y, 2) a single, or say a onefold relation, which can be represented by a single relation f(.;::, 3/, z)=0 ; or we may consider the coordinates expressed each of them as a given function of two variable parameters 1), g; the form 2.-_—f(.r, 3/) is a particular case of each of these modes of representation; in other words, we have in the first mode f (.r, y, 2) =2 —f(.I:, y), and in the second mode :0 =1), 3/ = q for the expression of two of the coordinates in terms of the parameters. In the case of a curve we have between the coordinates (.75, 2 , 2) a twofold relation: two equations j(.v, 3/, 2) =0, gb(.v, y, z)=0 give such a relation; 2'.e., the curve is here considered as the intersection of two surfaces (but the curve is not always the complete intersection of two sur- faces, and there are hence ditficulties); or, again, the co- ordinates may be given each of them as a function of a single variable parameter. The form 3/ = gbx, 2 = ipm, where two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation. 29. The remarks under plane geometry as to descriptive and metrical propositions, and as to the non-metrical char- acter of the method of coordinates when used for the proof of a descriptive proposition, apply also to solid geo- metry ; and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres. The proof is obtained from the consideration that S and S’ being each of them a function of the form
- 62 + 3/2 + 22 + /12: +123; + cz + d, the difference S — S’ is a
more linear function of the coordinates, and consequently that S —- S’ = 0 is the equation of the plane containing the circle of intersection of the two spheres S =0 and S’: O. ilfcf/'i'cI.(l T/U302‘!/. 30. The foundation in solid geometry of the metrical theory is in fact the before-mentioned theorem that if a E T R Y finite right line PQ be projected upon any other line 00' by lines perpendicular to 00', then the length of the pro- jection 1"Q' is equal to the length of PQ into the cosine of its inclination to l"Q'—or (in the form in which it is now convenient to state the theorem) the pei-pendicular distance P13’ of two parallel planes is equal to the inclined distance l’(3 into the cosine of the inclination. lleiice also the. algebraical sum of the projections of the sides of a closed polygon upon any line is=0; or, reversing the signs of certain sides and considering the polygon as made up of two broken lines each extending from the same initial to the same terminal point, the sum of the projec- tions of the one set of lines upon any line is equal to the sum of the projections of the other set of lines upon the same line. Vhen any of the lines are at right angles to the given line (or, what is the same thing, in a plane at right angles to the given line) the projections of these lines severally vanish. 31. Consider the skew quadrilateral QMNP, the sides QM, MN, N P being respectively parallel to the three rect- angular axes 0.1:, ()_i/, 02; let the lengths of these sides be 5, 77, C, and that of the side QP be = p; and let the cosiiies of the inclinations (or say the cosine-inclinations) of p to the three axes be a, [3, -y; t-hen projecting siicccssively on the three sides and on (JP we have ‘$7 7]: §=Pa7 P/3: P72 P = a5 + /37: + 7C , whence p‘-"=§2+77‘-’+ (2, which is the relation between a distance p and its projections jc‘, 77, C upon three rect- angular axes. And from the same equations we obtain a‘-’+[32+72= l, which is a relation connecting the ('usi1iu- inclinatioiis of a line to three rectangular‘ axes. Suppose we have through Q any other line QT, and let the cosine-incliiiatioiis of this to the axes be a’, B’, 7’, and 6 be its cosine-inelinatioii to QP; also let p be the length of the projection of QP upon QT; then projecting on QT we have 2) -= a'£+B'n+7'§, p8. And in the last equation substituting for g. 17, C their values pa, pB, p7 we find [soLii) .-Z'.-I.YTICAL. and 8 = ac/4 BB'+77', which is an expression for the mutual cosine-iiiclinatioii of two lines, the cosine-inclinations of which to the axes are a, B, 7 and l I I . ‘ 7 .. 2 ‘L’ 2_ a , B , 7 iespectively. e have of course a +B +7 »1, and a"-’ + 3'3 + ')2= 1; and hence also 1 — 8'3= a2+ B3 + 7'*’)(a"'l' + 3'2 + 7'3) — (aa' + BB’ + 77')? = _ BIT)‘: + (ya: _ 7Ia)2 + (aflr __ alB"‘3 ; so that the sine of the inclination can only be expressed as a sqiiare root. These formula: are the foundation of spherical trigonoinetry. T he Line, Plane, uml S1':Iirv)v'. 32. The foregoing formulae give at once the equations of these loci. For first, taking Q, to be a fixed point, coordinates (a, 12, c) and the cosine-incliiiations (a, B, 7) to be constant, then P will be a point in the line through Q in the direction thus dcternnned: or, taking (.7:, ji/, 2) for its coordinates, these will be the curn-nt co- ordinates of a point in the line. The values of §, 77, f “WI! =ll‘1‘- a: — (I, 3/ — I), 2- c, and we thus have ‘ 2 - c x—a u—b —=- ——=>, _y( P a. 73' which (omitting the last equation,_=p} are the equations of the line through the point (a, I), c), the cosine-inelinatioiis to the axes being a. B, 7, and these quantities being connected by the l'OlZlllnI1 a3+B'-'+f=1. This equation may be omitted, and then a, B, 7, instead of being equal, will only be proportional to the CO>)llC- inclinations. _ Using the last equation, and writing 9‘, 2/, 2 a+ap. 7I+Bp, 0+‘/P. these are expressions for the current coordinates in terms of a para- meter p, which is in fact the distance from the fixed point (a, I», c). It is easy to see that, if the coordinates (91, y, 2) are connected by
any two linear equations, these equations can always be brought