At any point p of the line we have, if pAq = 0, pBq= 4, .. (a) may be written k (Aq + Dq) = 1 (Dq − Bq), i.e. k. AD-l. BD, which by construction it does.] To find points on the locus represented by (1). With the points A and B as centres describe two circles S and T of radii meeting S in L and T in N; then the lines AL and BN intersect in a point, P, on the required locus; for or or AQ+QB = AB, AL cos 0+BN cos &= AB, if BAL is 0 and ABN is 6, which is the given equation. с If the line DC meet the curve in R and R, the angles RAB, RAB are the required values of 0, and the angles RBA, R,BA those of p. There is a precisely similar loop on the other side of AB. In the particular case in which ab the locus is the Magnetic Curve. (Prob. 168.) Take two points A and B such that AB =a+b; make 40 = a, Q37 - 5 sund draw OD perpendicular to AB; with A as centre and * radius describe a circle, and draw any radius AC meeting 0 in 4; inflect BJ = LC (J being on OD); then P, the point of mtersection of AC and BJ is a point on the locus represented by (1), the angles and 4 being ALO and BJO respectively. There is a precisely similar loop on the other side of AB. Again the equation cos 0=k cos & gives sin PAB = k.sin PBA o PB k. PÅ, i.e. P is the vertex of a triangle on a given base and with sides in a given ratio (Problem 17), i.e. the locus represented by the second equation is a circle whose diameter is the line joining the points which divide AB internally daxternally in the ratio 1: k; ie. AQ: QB :: 1: k :: AQ1 : Q1B. The values of and which satisfy both equations are those the points of intersection of this circle and the [Trace the loci y = sin x (harmonic curve) and y line through the origin): the values of x corresponding to their points of intersection are solutions.] [The intersections of the harmonic curve y = sin x and of the straight line y: = ax + b where a and b are constants.] 3. Solve the equation 2o = 5 sin 0. [The intersections of the equiangular spiral r = 2o and of the circle r 5 sin 0.] 4. Find and from the equations tann tan 0.. a cos 0 = b cos & + c where a, b, c and n are given constants. ..(1), (2), [The 2nd equation represents a locus identical with (1) of Problem 181, attention being paid to the usual conventions as to sign. The 1st equation represents a right line perpendicular to AB (fig. 185), the base of this locus, and meeting it in D so that AD=n. BD.] AB [Draw two lines AB, AC including an angle a, and mal. =a and AC = m. With centres B and C and radii = n and respectively describe arcs intersecting in D on the same side of BC as AB. Let CD meet AB in E. BED is the required value of and DBE that of p.] 6. Determine 0 from the equation a cos λ. cos (λ + 20) = c. cos (a + 0) where a, c, λ and a are given constants. [The locus represented by the right-hand side of the above equation is a circle of radius c, the origin (0) being the extremity of a diameter, and the initial line making an angle a therewith. To draw the locus represented by the left-hand side :—draw a line through the origin O making an angle λ with the initial line, and on it measure a length OL = a. Draw LN perpendicular to the initial line meeting it in N so that ON = a cos λ. With centre and radius ON describe a circle. From any point Q on this circle draw QM perpendicular to OL meeting it in M. Draw OP bisecting the angle NOQ and make OP = OM. P will be a point on the second locus, and any additional number of points may be similarly determined. Let the two loci intersect in X, and the angle between OX and the initial line is the required angle 0.] This equation defines the position of equilibrium of a uniform rectangular board resting in a vertical plane against two equally rough pegs in a horizontal line. THE END. CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. & SON, AT THE UNIVERSITY PRESS. Applied Mechanics: an Elementary General Introduction to A Treatise on Differential Equations. Integral Calculus, an Elementary Treatise on the; Founded on the Method of Rates or Fluxions. By WILLIAM WOOLSEY JOHNSON, Professor of Mathematics at the United States Naval Academy, Annopolis, Maryland. Demy 8vo. 88. Curve Tracing in Cartesian Co-ordinates. By the same Author. Crown 8vo. 4s. 6d. Differential Calculus, an Elementary Treatise on the; Founded on the Method of Rates or Fluxions. By JOHN MINOT RICE, Professor of Mathematics in the United States Navy, and WILLIAM WOOLSEY JOHNSON, Professor of Mathematics at the United States Naval Academy. Third Edition, Revised and Corrected. Demy 8vo. 16s. Abridged Edition, 88. A Treatise on the Calculus of Variations. Arranged with the purpose of Introducing, as well as Illustrating, its Principles to the Reader by means of Problems, and Designed to present in all Important Particulars a Complete View of the Present State of the Science. By LEWIS BUFFETT CARLL, A.M. Demy 8vo. 21s. A Treatise on the Dynamics of the System of Rigid Bodies. By EDWARD JOHN ROUTH, D.Sc., LL.D., F.R.S., Fellow of the University of London, Hon. Fellow of St Peter's College, Cambridge. With numerous Examples. Fourth and enlarged Edition. Two Vols. 8vo. Vol. I.-Elementary Parts. 14s. Vol. II.-The Advanced Parts. 14s. A Text Book of the Method of Least Squares. 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