. An elementary treatise on the differential and integral calculus. PT = OQT + eld, the two angles OPT aud OQTdiffer from each other by an infinitesimal, and thereforeOPT = OQT, and hence, reldtan OPT = -~, from (3), (4) sin OPT == sin OQP = §p = ^, from (1). (5)Hence,OT = polar subtangent = OP tan OPT = -jl, from (4). (6)OC = polar subnormal = OP tan OPC = OP cot OPT = g, from (4). (7) PT = polar tangent = VOP2 + OT2 = r i/l + . eli2 from (6). (8) PC = polar normal = VOP2 + OC* = y r2 4- ^~. eld2 from (7). (9) 180 EXAMPLES. OD = p = OP sin OPD = -p from (5) = ds See Prices Calculus, Vol. I, p
. An elementary treatise on the differential and integral calculus. PT = OQT + eld, the two angles OPT aud OQTdiffer from each other by an infinitesimal, and thereforeOPT = OQT, and hence, reldtan OPT = -~, from (3), (4) sin OPT == sin OQP = §p = ^, from (1). (5)Hence,OT = polar subtangent = OP tan OPT = -jl, from (4). (6)OC = polar subnormal = OP tan OPC = OP cot OPT = g, from (4). (7) PT = polar tangent = VOP2 + OT2 = r i/l + . eli2 from (6). (8) PC = polar normal = VOP2 + OC* = y r2 4- ^~. eld2 from (7). (9) 180 EXAMPLES. OD = p = OP sin OPD = -p from (5) = ds See Prices Calculus, Vol. I, p. 417. from (2). (10) EXAMPLES 1. The spiral of Archimedes, whose equation is r = ad.(Anal. Geom., Art. 160.) Here dO _ lmdr ~ a9 Subt. = -, from (6), Subn. == a, from (7). s> Tangent = r\ 1 4- -2, from (8), Normal = Vr2 + a3, from (9). /> = yV* + from (10). 2. The logarithmic spiral r = a1 Art. 163.) drHere — = ae log a =? r log a; (Anal. Geom., Subt = log « = m r, (where m is the modulus of the systemin which log a = 1). Subn. = * m mr 1/1 + V^2 + 1. Fig. 20 DV Normal = r2 + -^ | = (r» + r2 log2 a)i RECTILINEAR ASYMPTOTES. 181 Tan. OPT = r ^ = ^ ;ar log a which is a constant; and therefore the curve cuts everyradius-vector at the same angle, and hence it is called theEquiangular Spiral. If a = e, the Naperian base, we have, tan OPT = .— = 1, and .-. OPT = 45°,log e and OT = OP = r. 3. Find the subtangent, subnormal, and perpendicular inthe Lemniscate of Bernouilli, r2 = a2 cos 29. (, Art. 154.) Subtangent = -$—.—r^; ° a2 sm 20 2 CI Subnormal = — - sin 29r Perpendicular Vr4 + a* sin2 29 a 4. Find the subtangent and subnormal in the hyper-bolic* spiral r9 = a. (Anal. Geom., Art. 161.) r2 Subt. = — a ; Subn. = • a RECTILINEAR ASYMPTOTES. 103. A Rectilinear Asymptote is a line which iscontinually approaching a curve and becomes tangent to itat an infinite distance from the origin, and yet passeswithin a finite distance of the origin. To find whe
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