Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . ^ obtained. THE Lixrrs. 219. The spiral in which the radius vector is inversely proportional to the square root of theangle through which it has revolved is called the Lituu-s. This relation is shown bv the equation r = 7^1 also written 6 = - .ay 6 r When 0 = 0 we find r = oo. making the initial line an asymptote to the curve. In Fig. 127 take 0 Q as the initial line. 0 as the pole, a = 2, and our unit a three-inch line: then - = 1;.a THEORE
Practical engineering drawing and third angle projection, for students in scientific, technical and manual training schools and for ..draughtsmen .. . ^ obtained. THE Lixrrs. 219. The spiral in which the radius vector is inversely proportional to the square root of theangle through which it has revolved is called the Lituu-s. This relation is shown bv the equation r = 7^1 also written 6 = - .ay 6 r When 0 = 0 we find r = oo. making the initial line an asymptote to the curve. In Fig. 127 take 0 Q as the initial line. 0 as the pole, a = 2, and our unit a three-inch line: then - = 1;.a THEORETICAL AND PRACTICAL GRAPHICS. For e = 90° = TT (in circular measiu-e ) we have r= 0 M=l!-2 +. - For 6=1 we havethe radius 0 T making an angle of 57 ? 29 + with the initial line, and in length equal to - units, i. e., U. For 6=45°= -f (or ) r will be OR=V:i +. Then 0H = 0 R for in rotating to 0 H the radius vector passes over four 45° angles, and the radius must therefore be one-half what it was for the first 45° described. Similarlj^ 0 K =—^; OM — —^, etc.; this relationenabling the student to locate an_y number of To draw a tangent to the curve Ave emjoloy the relation tan ^=26, <^ being the angle madebj the tangent line with the radius vector, while 0 is the angular rotation of the latter, in circularmeasure. BRUSH TINTING AND SHADING. 79 CHAPTER VI. TINTIJJG —FLAT AND GRADUATED.— MASONEY, TILING, WOOD GRAINING, RIVER-BEDS ANDOTHER SECTIONS, WITH BRUSH ALONE OR IN COMBINED BRUSH AND LINE WORK. 220. Brush-work, with ink or colors, is either flat or graduated. The former gives the effect ofa flat surface parallel to the paper on which the drawing is made, while graded tints either showcun-ature, or—if indicating flat surfaces—represent them as inclined to the paper, , to the planeof projection. For either, the paper should be, as pre^dously stated (Arts. 41 and 44) cold-pressed andstretched. The surface to be tinted should not be ab
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