This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1905 edition. Excerpt: ...(Vei + ai, i), (Vej+bj, j), (VeK+ck, k), situated on screws whose axes i, j and k are mutually rectangular and which intersect at the extremity of the vector e. The corresponding pitches are of course a, b and c; the latent roots of the self-conjugate part of the function f. The pitch of the wrench (pX, X) and ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1905 edition. Excerpt: ...(Vei + ai, i), (Vej+bj, j), (VeK+ck, k), situated on screws whose axes i, j and k are mutually rectangular and which intersect at the extremity of the vector e. The corresponding pitches are of course a, b and c; the latent roots of the self-conjugate part of the function f. The pitch of the wrench (pX, X) and the vector perpendicular on its axis are respectively (Art. 99) p = SjX.X-zs = VjX.X-1; (vi.) thus p is the reciprocal of the square of the radius of a quadric and the vector zs terminates on the surface represented by sws+1=0 TM-because Vct'ct X and therefore zs = VVrsf'rs(Vrs(l'a)-1; and this surface is a quartic with three intersecting double lines--the axes of f. (Steiner's quartic surface.) When the origin is taken at the extremity of the vector e, the function f is self-conjugate. This point is the centre of the three-system of screws. In terms of the pitch p and the vector Ij from the centre to any point on the axis of a screw of the system, M = pX + V, X = 0X = f X (via) See Joly, Tram. R.I.A., Vol. xxx., Part xvi., and Vol. xxxii., Part viii. Art. 103. To give an example of applying quaternions to a problem in statics, consider the case of a chain lying on a smooth surface and acted on by any force. let be the force per unit mass, v the normal reaction per unit length, w the mass of the chain per unit length, and P the tension of the chain. For equilibrium of an infinitesimal element at the extremity Of P, d(PVdP) + wiTdP + vTdp = 0, Svdp = 0, (i.) the pull back at p being--PVdp and the pull forward &t p + dp being +PUdp + d(PUdp). When the length of the chain is Art. los. CHAIN LYING ON SURFACE. 167 taken as the independent variable (Art. 85, p. 133), this may be written P.p"+FP'+w +v=0, Sp'P" = 0, Sit'=0;...
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