The Philosophical magazine; a journal of theoretical, experimental and applied physics . ur lines in one plane, (<-) efg the symbols of the lines OP, OR, and hkl, uviv the symbols ofplanes parallel to the lines OQ, OS. Each of the angles POQ, ROQ, POS, ROS being supposedless than two right angles, sin POQ sin ROS, sin ROQ sin POSwill have the same signs except when one only of the lines OQ,OS lies in the angle POR. 29. Let h, 1, u, w be the orthogonal projections of the pointsH, L, U, W in the preceding figure, on a plane through O, per-pendicular to the intersection of the planes hkl, uvw.
The Philosophical magazine; a journal of theoretical, experimental and applied physics . ur lines in one plane, (<-) efg the symbols of the lines OP, OR, and hkl, uviv the symbols ofplanes parallel to the lines OQ, OS. Each of the angles POQ, ROQ, POS, ROS being supposedless than two right angles, sin POQ sin ROS, sin ROQ sin POSwill have the same signs except when one only of the lines OQ,OS lies in the angle POR. 29. Let h, 1, u, w be the orthogonal projections of the pointsH, L, U, W in the preceding figure, on a plane through O, per-pendicular to the intersection of the planes hkl, uvw. In thisplane draw lines perpendicular to Ou, hi, Ow, uw, meeting acircle described round the centre 0, in the points P, Q, R, , OS are respectively perpendicular to the planes hkl, , OR are respectively perpendicular to planes containingthe lines efg, pqr, parallel to the intersection of the planeshkl, uvw. By similar triangles, Oh_ _ Ou ^ 01_ _ 0M^ On~OU OL^OWTherefore sin PQ sin RS sin Ohl sin Own sin RQ sin PS__01 Ou~ Oh Ow Hence sin PQ sin RS sin Olh sin OuwOL OU oh ow. e/i + ik -I- g/ pw + qy + yw (t) sin RQ sin PS ph + qk +17 eu + h + gw where OP, OQ, OR. OS are perpendiculars to four planes inter-secting one another in parallel lines, efg, pqr the symbols oflines in the planes perpendicular to Of*, OR, and hkl, uvw thesymbols of the jilancs perpendicular to OQ, OS. Each of the arcs PQ, RQ,.PS, RS being supposed less thana semicircle, sin PQ sin/^^\ sin7?QsinPS will have the samesigns except when one only of the points Q, S lies in the arc PR. 30. The expressions for the indices of a plane when referredto new axes, are easily deducible from {p). Let efg, hkl, pqr be Phil. May. S. 4. Vol. 18. No. 117. July 1859. E no Mr. J. Cockle on the Theory of Equations the symbols of the straight lines OU, OV, OW meeting theplane X^v in A, M, N, and tlie plane uvw in U, V, AV. Bysubstituting A, N for H, L, and X, fx, v for h, k, I in (p), andthen writing M for N, and h, k,
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