A table conveniently exhibits these relations: In the elements of solid analytic geometry, it is shown that the determinant of the nine di rection cosines equals 1, i.e. I a, 11, Yr =1, a, Y2 a, y, and that each constituent is equal to its co factor. For example, It is also a theorem of the elements of analytic geome try of space, that if a straight line varies its direction infinitely little, say by an angle dw, and if 1, m, n are the direction cosines of the first position, and I+ dl, m+dm, n+dn are the direction cosines of the second position, that It has immediate applica tion in the two curvatures.
Taking over from the calculus the value of d.? and differentiating, (3) ds d's=dx crx+dy dly+ds The analytical side of the foregoing develop ment is now easily formulated. All equations which follow and which contain the variables x y, s and their differentials dx, dy, ds, dlx, d'y, ds, d's may be expressed in terms of the parameter u and its differential du by means of equations (2), the equations resulting from their differentiation, and equations (3).
One has immediately from the definitions of the elements at the point P(x, y, u) dx dy ds (4) a j - ds yi hence, for the equations of the tangent, c—s (5) =.-- dx dy ds and for the normal plane, (6) (5— x)dx (A—y)dy (C —s)ds O. The symbols e, r, C represent here and sub sequently the current co-ordinates of the points of line, plane, etc.
In determining the constants L, M, N, Q so that the plane Lf + Mn + NC +Q----0 passes through the point x, y, „s, and the infinitely near points x+dx, y+dy, z+dz, and x+2dx+ d'x, z+2ds+ez, one obtains (dy d'y)+ (ds d'x—dx d's) + (dxcPy—dy or, in putting for convenience (7) A=dy d'y, B=ds ex—dx C=dx cPx, the equation of the plane of osculation in the form (8) A (f — x) —y)— s) =-- O.
This at once gives A (9) a,f3, , f + + V + ± V442+B2+ C2' and for the equations of the binormal —x (10) B From the determinant of the direction co sines one has (11') As=ear—asyl, and hence, by virtue of (4) and (9), Bdz—Cdy Cdx—Adz ..
ds"V dsv Ady--Bdx 7,= ds 1,/ The equation of the plane of rectification is (12) (f—x) as+ (v—y) 1;,+ Y2=0, and the equations of the principal normal are (13) c s • sa Applying to curvature the theorem relative to the infinitely small change in direction of a right line, one has del = + + whence, in differentiating (4) and employing (3) in the reduction, 1 d (14) (curvature)' — and also (15) daa=. Bdz—Cdy Cdx—Adz I 01= f dy.= A ds' A comparison of (11), (14) and (15) gives dal dPi d yi (16) os=R """'"/ A ds ds ds When the arc s is the independent variable, that is, when u = s, equations (14) and (16) take simple forms, 1 ex d's ds' ds' ds' elsz = R y, = R — • dss Applying the same process to torsion dr' = dais -I- Oil dy,', and a differentiation of (9) gives Bds — Cdy Adsx+ Ms), Cdsz V A' + + Cdx — Ads Ada,: + + Cd's