An elementary course of infinitesimal calculus . Fig. 100. As 6 ranges from — oo to + oo, r ranges from 0 to oo,See Fig. 100. 140] SPECIAL CURVES. 367 Since, by Art. 110, we have clrjds = cos a, it appears that thelength of the curve, between the radii r^, r^, is /, — dr= (r^ — ri) sec .(3). 2°. The spiral of Archimedes is the curve described bya point which travels along a straight line with constantvelocity, whilst the line rotates with constant angularvelocity about a fixed point in it. In symbols,whenceif a = m/k. r = ut, 6 = nt,r = aO ?(4),. Fig. 101. Fig. 101 shews the curve. The dot
An elementary course of infinitesimal calculus . Fig. 100. As 6 ranges from — oo to + oo, r ranges from 0 to oo,See Fig. 100. 140] SPECIAL CURVES. 367 Since, by Art. 110, we have clrjds = cos a, it appears that thelength of the curve, between the radii r^, r^, is /, — dr= (r^ — ri) sec .(3). 2°. The spiral of Archimedes is the curve described bya point which travels along a straight line with constantvelocity, whilst the line rotates with constant angularvelocity about a fixed point in it. In symbols,whenceif a = m/k. r = ut, 6 = nt,r = aO ?(4),. Fig. 101. Fig. 101 shews the curve. The dotted branch correspondsto negative values of 6. Another mode of generation of this curve has been explainedin Art. 138. 368 INFINITESIMAL CALCtJLUS. [CH. IX 3°. The reciprocal spiral is defined by the equationr = aie (5). If y be the ordinate drawn to the initial line, we have sind f = r sin 6 = a 6 As 0 approaches the value zero, r becomes infinite, but yapproaches the finite limit a. Hence the line y = a is anasymptote. Y
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