1-^ THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prell S^k^ '1 ^■fv- WORKS PUBLISHED BY W. P. GRANT. CAMBRIDGE. THE POETICAL, with a Selection of the PROSE WORKS of THOMAS CHATTERTON ; numerous illustrative Notes; and facsimiles copied from the Originals in the British Museum, expressly for this Edition ; to which are added the Life of Chattekton, containing many new facts, and the History of the ROW- LEY CONTROVERSY, 2 vols. fcp. 8vo. chih, \5s.; large paper, 21.?. The scarcity of the edition of the Works of Chatterton, edited by Southey and Cottle, is a sufficient reason for the announcement of the present publication. The strange beauty of the Rowley fiction — the exquisite pathos and soaring imagination of Chatterton — and fhe peculiar character and dark late of that wonder- ful genius — alike interest the poet, the antiquary, and the philosopher. " The poems bear so many marks of superior genius, that they have deservedly excited the general attention of polite scholars, and are considered as the most remarkable productions in modern poetry. We have many instances of poetical eminence at an early age ; but neither Cowley, I\Iilton, nor Pope, ever produced anything, while they were boys, which can justly be compared to the poems of Chatterton. The learned antiquaries do not, indeed, dispute their ex- cellence. They extol it in the highest terms of applause. They raise their favourite Rowley to a rivalry with Homer ; but they make the very merit of the works an argument against the real author. Is it possible, say they, that a boy could produce compositions so beautiful and so mas- terly ? That a common boy should produce them is not possible ; but that they should be pro- duced by a boy of an extraordinary genius, such a genius as that of Homer and Shakspeare, such a genius as appears not above once in many centuries, though a prodigy, is such an one as by no means exceeds the bounds of rational credibility." — V. Knox. CAMBRIDGE UNIVERSITY MAGAZINE, No. XL Price 2s. 6rf.— Con- tents : — A Stroll in and about Oxford — Specimens of the German Anthology. No. III. — Plato and Mr. Sewell. — Queen Melior. Part III. — The Poets of England who have died young. No. V. — Byron in Greece — Withered Leaves — Visions of the Heroes of Poland. — My soul was dark and desolate— The Living Dramatists of England. No. IV.— T. N. Talfourd — Review : the Latin Poems com- monly attributed toWalter Mapes — University Intelligence — Classical Tripos Papers. No. XII. will be published on the 12th of November. 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GEOGRAPHY (The) of HERODOTUS, with Six MAPS. ito. cloth, 6s. ^''' ~ ^)' y' = - (« - ^) (!)• U ■ d The formula for x will be, indicating the limits, X dx dy J oJ 1,1 ndx dy J' Now I I xdxdy =\ {y - y') x dx J oJ y Jo = - I {(a" - x'f - (a - x)} X dx, by the equations (1), Cl J h > ,2 = - { - 3 («■ - '^■■/ - \ a^ + 3 ^^}j from x=0 to x = a, ^Wb. CENTRE OF GRAVITY. 11 Also dx chj = \ (y - y') dx = - {(«" - x^J - {a - x)] dx J uJ y' Jo ClJo aj X I, d - xy dx — (or - \ a') - {\ TTcr - i (f) = i (tt - 2) ab. Hence from (2), X 2 « IT - 2 Similarly we should efficiently get y - 3 TT — 2 (3) To find the centre of gravity of the area KSL (fig. G) of a parabola, of which S is the focus, and SK, SL, any two radii. Take ^S" as the origin of co-ordinates ; also, A being the vertex of the parabola, let ASx be the axis of x, and Sy at right angles to Sx the axis of y. Let SP = 7, Z-ASP = d, AS=m. Then for the position of the centre of gravity, if L.ASK=a, z.ASL = (5, r- cos (it -6) dO dr \ I ^ si^ (j^ ~ ^) ^^ ^^ J77- ' y wr- rdSdr r dd dr Now f ' r cos (tt - d) dO dr = i r" cos (tt - 0) dO = -Ir' cos dS ; J (I but, by the nature of the parabola, m cos g U fr COS hence r cos (ir - 9) d9 dr =- '^m' —j-— dS, J COS 2 " ^ r' COS (tt - 0) dO dr = -km j^ ^^^jq d9> cos a_ _ l-tan-|0 _L_ = Izl^!^ = (1 - tan^ 10) sec^ 10: "'^^^^S^^l+tan^ie cosne cosM0 ^ 12 CENTRE OF GRAVITY. hence rr r cos U - 6) cWdr = -l?n' P sec^" 10(1 -tan' ^0) sec' ^0 dB J aj J a = - im' (1 - tan* ^B) 2d tan ^9 j tania = - ff/f {tan i/3 - tan ^a Also j^f^ r dB dr^^( r dB i(tan^ i/3- - tan^ 1 a)}. ^ = hnf p dB J a COS* ^B run i^ = lm'l (1 f tan'if/) 2'TRE OF GRAVITY. (7) To find the centre of gravity of the portion of the area of the curve y = sin x, between x - Q and x = ir, (8) To find the centre of gra\-ity of the area intercepted between a straight line y = ^x and a parabola y' = 4m;r. _ 8m _ 2m Sect. 3. Solid of Revolution. Let a solid of revolution be generated by the rotation of a plane curve about the axis of x ; then the centre of gravity will be within the axis of x, its position being given by the forimda - ^ 1 1 xy fJ^ (Jy ^ /■^(y--y'-)r7a: ^ 'jjy dx dy f{f - y'-) dx y, y , being the limiting values of y for any assignable value of x ; if y' = 0, we have {xy^ dx \if dx ' If polar co-ordinates be adopted, which are frequently con- venient, the formula will be _ _ ///•' sin cos dQ dr "^ f f 7^ sin OdOdr ' the pole being taken at the origin of x, and being the angle of inclination of the radius vector r to the axis of x. (1) To find the centre of gravity of the segment of a sphere. The centre of the generating cii'cle being taken as the origin, its equation will be x' + y- = a' (1); and, c being the distance of the centre of the plane face of the segment from the origin, I xif dx X = Jc y^^^ (2); CENTRE OF GRAVITY. 15 but I xif dx = \ {a' - X') x clx, from (1), = ^ cfx' - J x^, from x = c to x - a, = :f a* - i «"c" + ^ c* = ^ («" - c^J; also y' f/a; = I (a" - 2;') dx = a^:^: - g^;^, from .^ = c to x ^ a, = § a^ - aV + I d" : hence from (2), _ ^ («' - c")" _ 3 (a + c7 3 „3 ~ 4 2a - 3a-c + c' ^ 2a + c If the segment become a semi-circle, then c= 0, and therefore Lucas Valerius ; De Centro Gravitatis SoUdorum, Lib. 11. Prop. 33, and Lib. iii. Prop. 31. Guldin; Centrobaryca, Lib. i. cap. 11, p. 130. Wallis; Opera, tom. i. p. 728. (2) To find the centre of gravity of the solid formed by the revolution of the sector of a circle about one of its extreme radii. Let /3 denote the angle between the extreme radii of the sector; then the centre of the circle being the origin of x, and a the radius, nr^ sin cos dQ dr ) (^^ ^ ~ r^ra Ujj I I r^ sin dO dr Jo J but rr r' sin 6 cos d9 dr = \a* f sin cos 6 d9 J J Jo = 1 a* I sin 20 d9 = -/e a* (1 - cos 2/3), Jo and r r r sin Q dB dr = W \ sin dd =la\\ - cos /3); J J Jo hence from (1) we have X = ^a , ~ ^Q^ ^ = la(\ + cos /3) = 5 a cos' i/3. 1 - cos p 10 CENTRE OF GRAVITY. We might equally well have integrated the numerator and denominator of (1), fu-st with respect to 9, and afterwards with respect to r. In the one order of integration, we conceive the sector to be made up of an infinite number of thin triangles, of wliich the centre of the chcle is the common vertex ; in the other order, the sector is conceived to be made up of an infinite number of infinitesunal rings, having the centre of the cu'cle as then* common centre. WaUis; Opera, torn. i. p. 728. (3) To find the centre of gravity of the solid generated by the revolution of the parabolic area ABC (fig. 9), about the tangent Ax at the vertex A, BC being at right angles to the axis Ay of the parabola. Taking Ax, Ay, as the axes of x, y, the equation to the curve will be X' = 4:f7iy. Let AC= a, BC = b; then I / xy clx dy \ {a' - y') x dx — J oJ „ Jo / ydxdy / (a' ~ y') dx J oJ V Jo (cr-if)dy 2nf y^ , from (2), J = 2m' - — ^^ = infa" = f, b. 2a^-la^ This is a case of a more general problem given by Carre, Mesure des Surfaces, Sec. p. 93. (4) To find the centre of gravity of the solid formed by the revolution of any parabola, of wliich the equation is ?/'"*» =r a"'.r". For any portion of the solid from a: = to x = b, _ m + Zn , X = b. 2m + An CENTKE OF GKAVITY. 17 (5) To find the centre of gravity of the frustum of a para- boloid. If a, h, be the radii of the less and of the greater ends, h the length of the frustum, and x the distance of the centre of gravity from the smaller end ; a' + 2lr X = Ui a" 4 ¥ (6) To find the centre of gravity of an hyperboloid. If the equation to the generating hyperbola be we shall have for the volume between x = and x- c, _ 8ac + 3c' ~ 4 (3a + c) ' Carre; Mesure des Surfaces, &c. p. 97. (7) ABC (fig. 10) is a portion of the area of a common parabola, where BC is at right angles to the axis Ax of the parabola; to find the centre of gra%-ity of the soHd gene- rated by the revolution of the area ABC about BC. Let BC = b; then, G being the centre of gravity, CG = ^b. Carre; II). p. 90. (8) AC, BC, (fig. 11) are the semiaxes of an hyperbola, AD being a portion of the cvurve intercepted by BD drawn parallel to CA ; to find the centre of gravity of the solid generated by the revolution of the area A CBD about CB. If BC=h, then G being the position of the centre of gravity '^^^BC, CG = ^b. Carre ; lb. p. 97. (9) To find the centre of gravity of the solid generated by the revolution about the axis of x of the ciu"ve corresponding to the equation l^ = (a-x)(^. 18 CENTRE OF GRAVITY. For any portion of the solid cut off by a section at a distance c from the origin, __ 140ff-e - 240q!C- + lOoc'' ^" 168«'- 280ac + 120c'' ' which for the whole solid becomes Carre ; Mesure des Surfaces, &c. p. 99. (10) To find the position of the centre of gi'avity of the volume included between the surfaces generated by the re- volution of two parabolas, y^ = Ix, y' = l' (a - x), roimd the axis of ^- _ , / + 21' x = \a l + l' Sect. IV. Any Solid. Let X, y, z, be the co-ordinates of any point whatever within any assigned solid ; let x, y, z, be the co-ordinates of the centre of gravity of this solid ; then - _ flf^ ^-^ '^h ^^ - fffi/ ^^ ^y ^^ - IH^ ^^^ ^^y ^^^ ^ ~ fff(^^ f^y (1^' ^ " fjjdxdydz ' ^ ^ ~jfjd^dfd^ ' where each of the ti'iple integrations is to be performed in accordance with the nature of the bounding surface of the solid. (1) To find the centre of gravity of a portion of the cone, of which the equation is y- + z- = /3V, which is contained between the planes of xz, xy, and a given plane parallel to that of yz. Let a be the length of the axis of the portion of the cone: then _jj^j^xdxdydz _ fj^Jjdxdydz j I f dx dy dz ' [" i ^r dx dy dz ' nj zdxdy dz I I I dxdydz J J Jo CENTRE OF GRAVITY. 19 Now I I j dx dy dz = \ \ z dx dy n » {^''x'-y-fdxdy J J = I 5 7r/3V dx = T^7rj3V. Also I I I X dx dy dz = I I a;z c?ir '^'- - \~n-. r''"'y''' . \ \ \ dx dy dz \ \ \ dx dy dz J -a J J z' J -a J J z where x' is taken to represent (a^ - off, and where the limits z, z, of the general value of z, are its values for any assigned values of x and y in the paraboloid and sphere respectively. Now C' i 1 I dx dy dz = dx dy (z - z) - dx dy {(a" -x^- xfj ia!'-:^- f)\; J z' -^^ hence j I dx dy dz = I dx dy {(cr - x"" - y^ (a' - a;^ - y^)} and therefore / "f r dx dy dz = f" dx UTria' -x')-— (a' - xj] J -aj J z- J -a 3a = i ira' -~ f (a' - xj dx = ] ira^ - ^ «"«' = Ij Tra^. OCl J -a Again, j y dx dydz = y dx dy {(a' - a;' - yj -~(a^-x^- f)], \ j ydxdydz= r ydxdy{(a'-x'-yy-—(d'-x'-f)} J J z' Jo 2ci 3 1 1 = dx{-i (a- -X- - yy - —- {a' - x^) y" + ~~ ^/], between lijnits, = dx{\{a'-xi-l-{a'-:tj]', hence I I \ ydxdydz = \l\ce-:irf dx-~r {a'-2a'3?^x')dx J -"-J J z' J -o 8aJ -a ,404 IStt- 16 . 120 L CENTRE OF GRAVITY. 21 Again, zdxdydz = l dx dy {z^ - z'^) = i dzdy {(a^- x^- tf) ^ (a - x^- y^J] , \ I z dx dy dz = l/J' dx dy {a^~x^-f- ^ {a^~:^J + l^^^a^ -x^)f -j-^f] = idxHa'-xy-'^(a'-xJ - —^ (a'~xj+ ~ (a'-x'f- -i-, (cr~x~f} between limits, a 1 k = \dx {(a" - a?y -„ (a- - xrj \, zdxdydz = \\ dx{{a^-x^J -—(a--x^J-\ J -aj J z' J -a 5a = U dx {a' -■ xj , (a' - xj dx J -a 15a' J -a = g Tra — jy Tra = ^^ ira . From the formulse for y and z then we have - T2o(157r- 16)a* IStt - 16 5 ^r,* = T,a. (3) AOC (fig. 12) is a right-angled triangle, O being the right angle ; A OBD is a rectangle, of which the plane is perpendicular to that of the triangle; from every point Ji in the line AC Si straight line HQ is drawn to meet BD in Q, in a plane at right angles to the areas of the rectangle and triangle ; to find the centre of gravity of the volume so generated. Let OAx, OBy, OCz, be taken as axes of x,y, z; from R draw RM at right angles to OA, join QM, and draw PN at right angles to QM; let OA = a, OB = h, OC=c; OM=x, 3IN= y, NP = z, z being the distance of any point in the line PN from the point N: then for the determination of the centre of gravity we have CENTRE OF GRAVITY. nl X (Jx (ly dz \ \ \ y dxdy dz _ )J u - _ J oJ i)J I I j dx dy dz j I j dx dy dz mz dz dy dz z = - .^ .^ ^ - • \ \ \ dx dy dz From the geometry it is e^icleut that a -- X h ~ y z = c ' — - — ; a hence v.e have \ \ X (a - x)(b - y) dx dy - J oJ ^ = -prp / (a - x)(b - y) dx dy J uJ I X (a - x) dx ^ 3 Jj_ = «^ = I a - -j-a la' "" j {a - X) dx ^ J u \ \ {a-x){b-y)ydxdy \ {h-y)ydy — J oJ _*'o ("lA {a-x)(b-i/)dxdy (i-y)dy s'' J oJ J IJ f-f" a-x h-y J J 2ab ff". ^.^ ^ . , c —^ dxdy I I {a-x)(b-y)dxdy J oJ a b J oJ c \a .\h' _ 2c ~ 2ab WTlb^ " "9 ■ (4) To find the centre of gravity of the portion of the sphere x' + y^ -^ z^ = a', which is cut off by three planes, x= 0, y = 0, z = 0. x = y = z = la. (5) To find the centre of gravity of a portion of the para- boloid y^ ^ ^2 ^ ^^^^ which is cut off by the three planes x = a, y = 0, 2 = 0. CENTRE OF GRAVITY. »3 If h be the radius of the section of the paraboloid made by the plane x = a, then _ ., _ _ 165 lOTT (6) To find the centre of gra\dty of a portion of the solid z' = xy, which is cut off by the five planes x=^, y = 0, z= 0, X = a, y = h. _ _ _ i 1 (7) To find the centre of gravity of the volume of the cylinder 1^ = 2ax - x^, which is cut ofi" between the two planes z = (5x, z = (5'x. x = %a, y=0, i = I (/3 + /3') a. (8) A solid is generated by a variable rectangle moving parallel to itself along an axis perpendicular to its plane through its centre ; one side of the rectangle varies as the distance from a fixed point in the axis, while half the other is the sine of a circular arc, of which this distance is the versed sine ; to determine the distance of the centre of gravity of the whole solid from the fixed point. The required distance is equal to four-fifths of the length of the axis. Sect. 5. A Plane Curve. Let X, y, be the co-ordinates of any point of a plane curve, and let ds denote an element of the length of the curve at that point; then x, y, denoting the co-ordinates of the centre of gravity of any assigned portion of the curve, — fx ds _ fy ds ""^Jdi' ^^7d7' the integrations being performed in accordance with the limits of the portion. The idea of the determination of the centres of gravity of curve Hues is due to La-Faille, a Flemish mathematician, by whom it was applied in the instances of portions of the circle and the ellipse, in a work entitled " De centro gravitatis partium circuli et ellipsis theoremata^'' published in the year 1632. The theorems of La-FaiUe were afterwards published in a somewhat 2-4 CENTRE OF GRAVITY. more elegant form, and with amplifications, by Guldin ; Centro- haryca, lib. i. cap. 4, 5, 6, 7. (1) To find the centre of gra\4ty of the arc of the curve y = sin X from x = to x = ir. From the equation to the ciu've we have ds^ = dx' + dy^ = (1 + cos^ x^ da? ; hence . , I sin a; (1 + cos* xf dx - lyds Jo I ( 1 + cos X) dx Jo Now by the ordinary processes of the integral calcidus. /: sin a; (1 + cos- xf dx = 2' + log (2'' + 1). Also, c denoting the length of the ciu've from x = to x = ir, c = (1 + cos' xf dx= 2^ I (1-2 sin' xf dx, Jo Jo an eUiptic function, of which cos lir is the modulus. Hence y is given by the equation cy = 2^ + log (2* 4 1). (2) To find the centre of gravity of any arc of a circle. Let the centre of the circle be taken as the origin of co- ordinates, and let the axis of x bisect the arc ; then, if a be the radius of the circle, c the chord of the arc, and s the length of the r arc. - ac _ ^ X = — , ?/ = 0. s -^ Guldin; Centrobaryca, Hb. i. cap. 5, p. 59. Wallis; Opera, tom. i. p. 712. (3) To find the centre of gravity of the arc of a semicycloid. The equation to the curve being X - y = a vers"^ - + (2ax - x^f , a we shall have AVallis ; Opera, tom. i. p. 520. - 2 « y = (^ _ 4) «_ CENTRE OF GRAVITY, 25 (4) To find the centre of gravity of the arc of a catenary X X y = ha {e'^ + e "), cut off by any assigned double ordinate. If 2s be the whole length of the intercepted arc, _ „ - ax + sy X = Q, y = ^ . (5) To find the centre of gravity of the arc of a parabola y~ = 4:mx, cut off by the latus rectum. _ , 3.2^ - loR- (\ + 2^) - ^ 2^ + log (1 + 2^) (6) To find the centre of gravity of the semi-arc of a loop of the Lemniscata of James Bernoulli. If the axis of the loop be taken as the axis of x, the node being the origin; then, a being the length of the axis and / of the semi-arc, _ «' _ (2^ - 1) a' ^ = — > y = — i — • 27 2n Sect. 6. Curve of Double Curvature. The formulae for the determination of the centre of gravity of a curve of double curvature, are _ _ (x (Is _ (y ds - _ f^ ds ""^'Jdi' ^^"7^' ^~~Td^' where x, y, z, are the co-ordinates of any point in the curve, and ds an element of the arc at that point : the limits of the in- tegrations will depend upon the positions of the ends of that portion of the curve of which the centre of gravity is required. Ex. To find the centre of gravity of the Helix. The equations to the curve are 2 2' I -1 "^ X + y =- a', z = cos - ; ^ a 26 CENTRE OF GRAVITY. and for the centre of gra^dty of any length of the curve, beginning at the origin of co-ordinates, - hii - b(a - x) _ , ^ = — , y = , z = hz. z z Sect. 7. Surface of Revolution. Let X, ij, be the co-ordinates of any point of a curve, by the revokition of which about the axis of x a surface is supposed to be generated ; then, if ds denote an element of the generating curve at the point, we have for the position of the centre of gravity of the siu'face of revolution in the axis of x, - fxy ds ^ (y ds the integrations being performed between limits depending upon the magnitude of the siu'face. (1) To find the centre of gravity of the surface of a segment of a sphere. If the equation to the generating cii'cle be y = {2ax - x'f, we shall have dy = dx, (2ax - x'f and therefore ds' = dx- + dy" = dx"" = ^^ , or y ds = a dx ; 2ax - X- y "^ hence for any segment, of which the limiting abscissa is c, ax dx , , /: r a dx c (2) To find the centre of gravity of the surface of a cone. Let the equation to the generating straight line be y = ax then, c being the length of the axis of the cone, x = lc. Guldin; Centrobaryca, lib. i. cap. 10, prop. 3. CENTRE OF GRAVITY. 27 (3) To find the centre of gravity of the surface generated by the revolution of a semicycloid about its axis. The equation to the curve being X - y = a vers"' - + {2ax - x^f , we shall have 157r - 8 _ 2 a ^ Stt - 4 (4) To find the centre of gravity of the surface generated by the revolution of the parabola ''/ = 4m:?; about the axis of x. X = l(3x - 2m). Sect. 8. Any Surface. Let X, y, z, be the co-ordinates of any point of a sur- face referred to three rectangular axes ; and let -^ = », dx dz — = 2> then for the centre of gravity of any portion of the dy surface _ ^ //•^^(i ^ f ^ q ~j dx dy X - ' 1 > //(I -^ P' ^ „, Di,, D^, denote these distances; then by Art. (1) we have T) - r^ D - r^ D =r^ where S denotes the spherical area ABC, and S^, iS^, S^., its projections upon the three great circles at right angles to OA, OB, OC. Now it is evident that the projections of the spherical area ABC, and of the sector BOC, upon the great circle which is at right angles to OA, are identically the same, and therefore, if the arc Aa be drawn at right angles to BC, we have S„ = area of sector BOC y. cos ( \1 r TT 9 . Aa TT 2 • T> ■ = ar' sm — = ar sin B . sin c : 360 r 360 30 CENTRE OF GRAVITY but S= — iA + B+C- 180); 180 ^ , -^ , a sin B sin c hence £>■ = kr -. — -, — " ^ ^ + j8+ C- 180 c,. ., , T-, , h sin Csin a oimilaiiy, X*^ = i *- />. = ^, r ^ + 5+ C- 180 c sin ^ sin b Z4TB + C - Tso If we desire to determine the position of the centre of gravity of the triangle by means of three rectangular co-ordinates x,y,z , let the plane of the side c be taken as the plane of x and i/, and let the radius OA be taken to coincide with the axis of x. Then from the preceding results we have at once _ , a sin B sin c ^ _ \ ^ ~ ^ cos A - a cos B ^ ^ '^^' A + B + C - 180 ' ''~''^'~~A^B'+C-180 ■ Again, let the great circle, of which i?C is an arc, meet the plane of x, z, in the point D, as in fig. 14 ; join A and D by an arc of a great circle. Then clearly the projection of the spherical triangle ABC upon the plane of x and z, is equal to the difference of the projections of the sectors AOC, BOC, upon this plane, and therefore to the expression r'^b cos CAD r^a cos D 180 180 = (b sin A - a sin B cos c) : 180^ ^' hence, by the principles of Art. (1), we have J b sin A - a sin B cos c ^"' A^ B^- C-\80 ■ (3) The general formula _ jjz{l+p^ + qidxdy furnishes us with the following general proposition : — " Upon the surface {A), generated by the revolution of the curve of equilibrium of a homogeneous catenary about the CENTRE OF GRAVITY. 31 vertical line which passes through its lowest point, trace arbi- trarily a perimeter enclosing a portion S of the surface ; project this perimeter upon a horizontal plane which intersects the axis of revolution at a distance c below the lowest point of the surface, where c is equal to the horizontal tension of the ca- tenary divided by the mass of a unit of its length ; let V be the volume contained between the surface S, its projection, and the cylindrical surface formed by the perpendiculars from the perimeter of S upon the plane of projection. Then the altitude of the centre of gravity of S above this plane will be double of that of the centre of gravity of V." In fact, the plane touching the surface (A) in a point situated at an altitude z above the plane of projection, which we shall take for the plane of x and y, makes with this plane an angle, c of which the cosine is - ; and therefore we have the equation z (i+p' + qi--; c hence, by the formula for z, we obtain - ^ //^' ^^ % _ ffz dx dy But calling z the altitude of the centre of gravity of V above the same plane, we have ^_ jj\zzdxd.y - Jfzdxdy and the limits of the integrations bemg the same in the expres- sions for z and z, we see that i = 2z. The property expressed by the partial differential equation (1 +p' + qy c being common to all the surfaces which can be generated by the surface {A) when it moves in such a manner that its axis always remains vertical, and that one of its points describes a plane curve traced arbitrarily upon a horizontal plane, the proposition which we have demonstrated is susceptible of the same exten- sion as that of (1). 32 CENTRE OF GRAVITY. The illustrations of the general formulae for the determination of the centre of gravity of any surface, which we have given in (1), (2), (3), are extracted from a memoir by Professor Giulio, of Turin, which may be seen in Liou\dlle's Journal de MatJie- matiques, tom. iv. p. 386. (4) To find the centre of gravity of the surface of a cone y- + 2;' = /3V-, intercepted by the planes x = Q, x = a, y =0, ^ = 0. - 2 - a /3« x^ia, y=.z = l-!-~ . Sect. 9. Heterogeneous Bodies. In the preceding sections we have computed the centres of gravity of various classes of homogeneous bodies ; we will now give a few examples of the determination of the centre of gravity when the density is variable. (1) To find the centre of gravity of the surface of a hemi- sphere, when the density of each point in the surface varies as its perpendicular distance from the circular base of the hemi- sphere. Let the equation to the quadi-antal arc, by the revolutiou of which the hemispherical sm-face may be generated, be x^ + 1/ = a"" (1), the axis of x being the axis of revolution. The area of the strip of the surface which is generated by the element ds of the arc, wiU be equal to 27r y ds ; and, if p be its density, its mass will be equal to 2Trpy ds : hence - _f27rpy ds . x _ fp xy ds j2irp yds ~ fpyds ' / ^ 1/ ds but p (X x; hence x = • . ; / xy ds and therefore, since from (1) it is easily seen that y ds = a dx, CENTRE OF GRAVITY, 1 x^dx , 3 we have x = -4 = ^—„ = \a /: xdx ' (2) To find the centre of gravity of a physical line, the density of which at any point varies as the n^"^ power of its distance from a given point in the line produced. Let a, h, be the distances of the given point from the two ex- tremities, and X its distance from the centre of gravity of the physical line ; then _ n+ I b"'^ - a"*"" X = w + 2 }f*^ - a"^^ ' (3) To find the centre of gravity of a quadrant of a circle, the density at any point of which varies as the n^^ power of its distance from the centre. Let a denote the radius of the circle, and x, y, the co- ordinates of the centre of gravity of the quadi'ant referred to its two extreme radii as axes ; then _ n + 2 la - X = . — = y. Sect. 10. Centre of Parallel Forces. When any number of parallel forces act on a system of rigidly connected points, they generally have a single resultant acting on a point of which the position is invariable while their common direction is changed in every possible way. This point is called the Centre of the Parallel Forces : the Centre of Gravity of a body is a particular case of this. Let x, y, z, denote the co-ordinates of the point of appHcation of any force P of the system referred to any axes, rectangular or obliqvie ; and let x, y, z, be the co-ordinates of the Centre of Parallel Forces. Then, 72 representing the resultant, ^ _,.„, _ ^(Px) - ^(Py) _ ^(Pz) 34 CENTRE OF GRAVITY. Whenever S(P) is equal to zero, these formulae cease to be appKcable, there being in this case no single resiJtant; the forces will be reducible to a resultant couple. For the complete development of the theory of Statical Couples, the reader is referred to Poinsot's beautiful work entitled EUmens de Statique. The formulae for x, y, z, were first given by Varignon, in the 31emoires de VAcadlmie des Sciences de Paris for the year IT 14. (1) Thi'ee parallel forces acting at the angular points A,B, C, of a plane triangle, are respectively proportional to the opposite sides a, h, c ; to find the distance of the centre of parallel forces from A. Produce ^^, AC, indefinitely to points x, y, and let Ax, Ay, be taken as co-ordinate axes. Let fxa, fxb, fic, be the forces applied at A, B, C, where a, h, c, denote the opposite sides of the triangle. The co- ordinates of the points of application of these thi-ee forces are 0, ; c, ; Q,h; hence by the two first of the formulae (B) we have - c . fih be y fxa + nh + fxc a + b + c' J . fic be fxa + fib + fxc a + b + c Let r be the distance of the centre of parallel forces from A ; then r' = ^* + y" + 2xy cos A = 2x' (1 + cos A) = 4i;' cos'^ ^ A, and therefore r = 2x cos lA = 2bc cos ^. a + b + c (2) Three parallel forces P, Q, R, act at the angles A, B, C, of a given triangle, and are to each other as the reciprocals of the opposite sides a,b, c; to determine the distance of their centre from A. -D . , J. . {b'+ 2bVcosA + cJ iiequired distance = a = , — . ab + ac + be CENTRE OF GRAVITY. 35 (3) At the corners B, C, D, of a quadrilateral pyramid ABCD, three parallel forces P, Q, R, are applied; to find the distance of their centre from the corner A. Let AB=h, AC=c, AD = cl; ^BAC={h, c), ^BAD=(h,d), Z. CAD ^{c, d)', r = the required distance ; then r\P + Q + Rf = P-5V Q-c' + R'cP + 2PQhc cos {b, c) + 2PRbd cos {h, d) + 2QRcd cos (c, r?). (4) At three fixed points (a, b), {a, b'), (a", h'), in the plane of X, y, are applied three parallel forces p,]^ ,p ; supposing the magnitude of p to vary in every possible way, to find the locus of the centre of parallel forces. The locus will be a straight line of which the equation is {ap + a'p)b" + {(«" - a)p + {a - ci^jp'} y = {bp + h'j)') a + 1(5" - b)p + (b" - b')p'} x. Sect. 11. 27*6 Properties of Pappus. I. If any plane area revolve about any axis in its own plane through any assigned angle, the volume of the surface generated by the motion of the area will be equal to a prism, of which the base is equal to the revolving area, and the altitude to the length of the path described by the centre of gravity of the area during its revolution. II. If any plane area revolve through any angle about any axis in its own plane, the area of the surface generated by its perimeter will be equal to a rectangle, of which one side is the length of the perimeter, and the other the length of the path de- scribed by the centre of gravity of the perimeter. The enunciation of these properties, which are generally called Guldin's properties, is due to Pappus,^ and may be seen ' The words of Pappus in the Latin translation are : " Perfectorum utrorumque ordinum proportio composita est ex proportione amphismatum, et rectarum li- nearum similiter ad axes ductarum a punctis, quse in ipsis gravitatis centra sunt. Imperfectorum autem proportio composita est ex proportione amphismatum, et circumferentiarum a punctis quse in ipsis sunt centra gravitatis, factarum, &c." In the former case he is alluding to those solids which are formed by the entire revolution of the generating figures through StiO"; in the latter, to those which are formed by revolution through any smaller angle. D 2 36 CENTRE OF GRAVITY. at the end of the Preface to the seventh book of his Mathemati- cal Collections, of which the first edition appeared in the form of a Latin translation in the year 1588. They were afterwards published, with vaiious applications, by Gnldin, in his treatise De Centra Gratitatis, lib. 2 and 3, which appeared for the first time in the year 1635. Cavalieri,^ in reply to objections ad- vanced by Guldin against his method of indi\"isibles, gave a demonstration of these properties by tliis method ; stating like- wise, in allusion to Guldiu's claims as a discoverer, that they had been communicated to him, long before the pubhcation of Guldin's work, by a pupil of his, Antonio Eoccha. Elegant demonstrations of these properties were given also by Varignon in the Memoires de V Acadtmie des Sciences de Paris for the year 1714, p. TT. (1) From any point P (fig. 15) in a parabola, is drawn a straight line PM at right angles to the axis, and meeting it in the point 31; to find the content of the solid generated by the complete revolution of the area APM about PM. Let A3I=x, P3I= y ; V= the required volume; and x = the distance of the centre of graA-ity of the area AP3I from P3I. Then the whole path described by the centre of gi*a-v-ity will be equal to 2irx; hence, by (I.), V= Ittx X area of PAM: but X = %x, and area of PAM = | a-y ; and therefore V= 27r . 5 :r . § ary = -^ irx-j/. Complete the parallelogram 3IPmA; then the area of this parallelogram will be equal to xy, and the distance of its centre of gravity from PM -odll be equal to \x. Conceive this pa- rallelogram to make an entire revolution about P3I; then the path of its centre of gravity will be equal to 2ir .\x = TTX; and therefore, if U denote the volume of the cyhnder which is generated by the revolution, U = TTX .xy = irary. Hence U : V :: 15 : 8. This is one of the problems proposed in Kepler's Stereometria. Guldinus; De Centro Gravitatis, lib. ii. cap. 12, prop. 6. ' Exercitationes Geometries Sex, Exercit. 1 & 2 ; Bononise, 1647. CENTRE OF GRAVITY. 37 (2) To find the surface of a sphere. Let BAb (fig. 16) be a semicircle, by the revolution of which about the diameter BCb the sphere is generated. Let CA be at right angles to Bb, C being the centre of the circle, and let G be the centre of gravity of the semicircular arc BAb. Let CA = a, sui'face required = S; then, by (IL), 2TrCG .arc BAb = S; 2a but CG = — , and arc BAb = ira ; hence TT aS = 27r . — . Tra = 47ra". TT Now TTor is the area of a great circle of the sphere ; and thus we find that the whole surface of a sphere is four times as great as that of one of its great cii'cles. This proposition was first proved by Archimedes, Uepl 2^a/pae rat KvXivSpov, Bt/3/\. A, TTpora. A; and afterwards, according to the method which we have given, by Guldin, De Centro Gravitatis, lib. iv. cap. 1, prop 7. (3) To find the volume and the surface of the solid ring generated by the complete revolution of a circle about any external line in its own plane. Let b be the distance of the centre of the circle from the axis of revolution, and a the radius of the circle ; then volume = l-KCi'b, and surface = 47r^aJ. (4) To find the volume of the solid ring generated by the revolution of an ellipse about an external axis in its own plane through an angle of 180°. If a, b, be the semiaxes of the ellipse, and c the distance of its centre from the axis of rotation, then volume = Tt^abc. (5) To find the volume generated by the revolution through a given angle of a portion APM (fig. 15) of a parabola about a tangent at its vertex A, PM being parallel to the tangent, and AM at right angles to it. 38 CENTKE OF GRAVITY. If AM= X, PM^ y, and /3 be the angle through which the revoUition takes place ; then volume = 5 )3.f'y- (6) To find the volume and the surface of the solid generated by the complete revolution of a cycloid about its axis. If a be the radius of the generating circle, volume = wa^ Q tt^ - |), sui-face = 87ra" (tt - g). (7) To find the volume and the surface of the solid generated by the complete revolution of a cycloid about its base. Volume = 5 ir~a^, sm-face = g^ ira^. (8) To find the content of the sohd generated by the complete revolution of a right-angled triangle about its hypothenuse. If a, b, denote the two sides of the triangle, the content will be equal to 3 (a- + h-j ( 39 ) CHAPTER II. EQUILIBRIUM OF A PARTICLE. Let P denote any one of a system of forces acting on a particle; and let a, (5, y, be the angles which the direction of this force makes with any three proposed straight lines, no two of which are parallel; then the sufficient and necessary conditions for the equilibrium of the particle are expressed by the three following equations, S (P cos a)=0, S (P cos i3) = 0, S (P cos y) = 0, where the S represents the summation of all such quantities as P cos a, P cos j3, P cos y, for aU the different forces of the system ; or the algebraical sum of the resolved parts of all the forces of the system estimated parallel to each of the three straight lines must be equal to zero. If all the forces acting on the particle lie within a single plane, then two of the thi'ee straight lines being taken in this plane, the three equations of equilibrium will evidently be reduced to two. The conditions for the equihbrium of a particle acted on by oblique forces, appear to have been first distinctly conceived by Stevin of Bruges.' He establishes by reasoning, which al- though indirect is satisfactory and ingenious, the ratio which the weight of a particle supported on an inclined plane bears to the force by which it is sustained, the force being supposed to act along the plane. He then announces generally, without however supplying a corresponding extension of demonstration, that the condition of equilibrium of any three forces acting on a particle, consists in the proportionality of the forces to the sides ' Beghinselen der Waaghconst, 1586. i. Lirre de la Statique, prop. 19. 40 EQUILIBRIUM OF A PARTICLE. of a triangle to which they are parallel. The first rigorous demonstration of Ste^in's theorem in its general form, was obtained by RobervaP from the nature of the lever. The idea of a triangle of equibbriuni had occurred indeed somewhat earHer to Michael Varro," of Geneva, in application to the equihbrium of forces acting on the sides of a right-angled- triangular wedge : it does not appear, however, that Varro's notion was based upon any very distinct conception of the nature of statical pressure. The Principle of the Parallelogram of Forces, which is in fact a mere modification of Stevin's theorem, was announced almost simultaneously by Newton^ and Varignon;* by whom it was inferred from the consideration of the composition of motions. In the same year was pubhshed by Land, in a Httle treatise entitled Nouvelle maniere de de- mo7itrer les principaux TJieoremes des eUmens des MecaiiiqueSy a theorem in which it is asserted, that if thi-ee forces P, Q, R, keep "a particle at rest, then P : Q : i2 : : sin (Q, i?) : sin (P, R) : sin (P, Q), where (Q, R), (P, R), (P, Q), denote the angles between the directions of Q and R, P and R, P and Q, respectively. The virtual coincidence of this theorem with the Principle of the Parallelogram of Forces, subjected Lami to the imputation of plagiarism, an aspersion cast upon him by the author of the Histoire des Outrages des Savans, (April 1688). Lami combatted this insinuation in a letter published in the Journal des Savans, (Sept. 13, 1688), to which the Jom-nalist replied in the following December, when the controversy appears to have tenninated. The fii'st unexceptionable demonstration of the Parallelogram of Forces on pure statical principles, wdthout the introduction of the idea of motion, M'as given by Daniel Bernoulli.' Many other proofs of the proposition have been ' Traite de Mecaniqiie, printed in 1636, in the Harmonie Universelle de Mersenne, and in a work also by Mersenne, entitled Cogitata Physico-Maihemaiica, published in 1644. * Tractatus de Motu, 1584. ^ Principia, lex iii. cor. 2, 1687. * Projet de la Nouvelle Mecanique, 1687. * Comment. Petrop., torn. i. p. 126, 1726. EQUILIBRIUM OF A PARTICLE. 41 since given. Eighteen demonstrations have been collected and examined by Jacobi^, by the following authors : D. Bernoulli, 1726; Lambert, 1771; Scarella, 1756; Venini, 1764; Araldi, 1806; Wachter, 1815; Kcestner, Marini, Eytelwein, Salimbeni, Duchayla; two different proofs by Foncenex, 1760; three by D'Alembert ; and those of Laplace and Poisson. Sect. 1. No Friction. 1. P and W(^g. 17) are two heavy particles; Wis attached to the end of a fine thread, and P is suspended from a fixed point C of the tlu-ead ; the tlii-ead has one extremity attached to a fixed point A, and passes tlu-ough a smooth small ring at B in the same horizontal line with A ; to find the ratio between P and TF, that the vertical line thi-ough C may bisect AB in D. From the supposition it is evident that z.ACD=LBCD; let each of these angles be denoted by : let 7" = the tension of the string CA; CA = h, AB = a; the ring B being perfectly smooth, I^F" will be the tension of the string BC. Hence for the equilibrimn of the point C we have, resolving vertically the forces which act on it, (r+ TF)cos B = P, and resolving horizontally, rsin0= TFsine, or T= TV; p hence 2TFcos0 = P, cos = —--.... (1); 1W but from the geometry we see that 5 sin = i a, sin = —7 (2). 26 Squaring the equations (1) and (2), and adding the resulting equations, we have _ P^ ^ P^ _ 45' - o^ ' Whewell's Philosophij of the luduclive Sciences, vol. I. p. 197. EQriLTBRlUM OF A PAKTICLE. and therefore p (4J2 _ ^y W h whicli determines the required ratio. 2. A particle P (fig. 1 8) is placed on the surface of a smooth prolate spheroid, and is attracted towards the foci S and H with forces varying as SP"" and HP'"'; to find the position of equilibrium. Draw a tangent KPL at the point P in the plane passing through the three points S, H, P ; let fi, fi , be the absolute forces towards S, H; SP = r, HP = r'. Then for the equi- librium of the particle we have, resolving forces parallel to the Hue KPL, fjL . f" cos Z.SPK = fir""' cos Z. HPL ; but L.SPK = L.HPL, by the nattu*e of elHpses; hence also 2a denoting the axis of the spheroid, 2a = r + r' ; hence for the determination of r and r we have the two equations /nr'" = fx (2a - r)'"', fi (2a - rj" = /n'r''"'. S. Two weights ?}i, m, are attached to the points 0, 0', (fig. 19) of a string AOO'A', suspended from two tacks at A and A' in the same horizontal line ; to find the positions of the points that their vertical distances from the horizontal line through A and A' may have given equal values. Draw Oi:, O'E', vertical ; let 0^ = a = O'E', A A = h, c = the length of the string; Z.^ 0^=0, ^A'0'E=6'; r= the tension of the string 00'. Then for the equilibrium of we have, by Lami's Principle, T sinfTr-fl) , a m sm (jTT + u) and for the equihbrium of O', ?n' sin Qtr + 0') ^ ^ -7f = -7-^7 — w{ = cot e. 1 sm (tt - a) From these two equations w'e get m tan B = m tan 0' (1). EQUILIBRIUM OF A PARTICLE. 43 Again^ from the geometry, EE' = AA' -AE-A'E' = b - a (tan + tan 6') ; but we have also, from the geometry, EE' = 00' = c-AO-A'0' = c - a (sec 6 + sec 6') ; hence a (sec - tan 6 + sec B' - tan 0') = c - b . . . . (2). From the equations (1) and (2) the values of 9, 9', are to be determined, and then, EO and E'O' being given, AO, A'O' will be known. Diarian Repository, p. 627. 4. To determine that point in the axis of a hemispherical body, the particles of which attract inversely as the square of the distance, where a corpuscle must be placed so as to remain in equilibrium by the equal and contrary action of the matter of the hemisphere surrounding it. Let CA (fig. 20) be the axis of the hemisphere, DCD' a diameter of its base, and the required position of the cor- puscle ; DAD' the intersection of the plane thi'ough CA, DCD, with the surface of the hemisphere; di-aw BOB' at right angles to CA, join OD ; take any points P, p, in the arcs AB, BD, join PO, po, and draw P3I, pm, at right angles to CA. Let CA = a= CD, CO - c, OB = b, OD = b', OP = r, 031= X, P3I--y, Op = r', Om = x', ptn = y'; ^u = the absolute attraction of a unit of mass of the hemisphere, and p = its density ; A = the attraction of the portion BAB' of the hemi- sphere on the corpuscle, and B of the portion BDB'D. The attraction of a thin slice of the hemisphere at right angles to its axis at the point 31, and having a tliickness dx, will be / x^ 27rjup dx ll -- as may be seen in elementary treatises on attraction ; hence 44 EQUILIBRIUM OF A PARTICLE. A = a c-r^-f^....(l); Jo r 2TTfip similarly we have B ==c-r^-^ (2). Jo r 2irfxp Now from the geometry we see that r^ = X' + if' = x^ ■}- a^ - {x + cf = a' - c^ - lex = 5^ - 2cx, hence 1cx = h^ - r, cdx = - r clr, - , „ xdx b' - /•* , and therefore = — dr ; r 2c' hence from (1), it being observed that r is equal to a- c, b, when X is equal to a - c, 0, we have A = a - c + ±j--'(,'-,'),r....(S). 2irnp Again, from the geometry, r'~ = x"' + y' = X' + a^ -{c - x'J = c^ - & -f 2cx = i^ + 2cx\ hence 2cx = r' - b^, cdx = r'dr, , , f, xdx b' - r' , , and thereiore — -- = dr : r 2c' hence from (2), since r' is equal to b', b, when x is equal to c, 0, B 2Trnp but it is evident that = c + i^J/^'-'-')*'^ dr r {b' - r") dr' = f (Jr - r) B 1 r*' hence = c + — ^ (h' - r') dr . . . . (4). 27r^|0 2c' ji ^ ^ ^ But since the corpuscle is in eqidlibrium we must have A = B, and therefore by (3) and (4), a - c 4 2c- L^j--\i'-r) tt^ • ^ P cos £ = jjlR + TV sm a ; and resolving forces at right angles to the plane, P sin 6 + i? = W cos a. Eliminating R between these two equations, we get P (cos E -h ^ sin f) = W{fx cos a + sin a) ... . (1). Let ^ be the least elevation of the plane for the particle without support to slide down it, then tan

^ = - — 7 Tx • 2 sm a sin a sin {a + a) sm {a + a) (12) A beam AB (fig. 34) leans against a smooth vertical prop CD, the end A being prevented from sliding along the horizontal plane AD by a string AD fastened at i>; to find the tension of the string. Let G be the centre of gra-\aty of the beam ; AG = a, CD = b, AD= c, W = the weight of the beam, T = the tension; then T = W. {1/ + 4 (13) A uniform rigid rod AB (fig. 28) rests upon a fixed point E, while its lower end A presses against a vertical line FF'; a weight P is suspended from the extremity B ; to find its position of equihbrium. Let 1V= the weight of the rod, b = the perpendicular distance of F from the line FF', AE = x, a = the length of the rod ; then / ,, P -f i W X = ab P+ W Fontana, Memorie della Societa Italiana, 1802, p. 630. If we suppose W = o, then we shall have x = (ab^J , whatever be the magnitude of P. This problem is discussed by Euler, Acad, des Sciences de Berlin, tom. vii. p. 196, in illustration of Maupertuis' Principle of Rest. (14) A sphere, of which C is the centre, is supported on an inclined plane AB by a string CB which is horizontal ; to find the tension of CB. If W be the weight of the sphere, and a the inclination of the plane to the horizon, tension of string = Tl^tan a. (15) A uniform rod of given length rests against a peg at the focus of a parabola, its lower extremity being supported on the curve ; to determine the angle wliich it makes with the axis of the parabola which is vertical. EQUILIBRIUM OF A SINGLE BODY. 61 If a be the length of the rod, and 4m the latus rectum of the parabola; then -1 A'^V requh'ed angle = 2 cos | — j^ (16) A weight TF hangs from a rod BC, (fig. 35), which rests on a fulcrum at B, and is supported by a string DA at right angles to the rod, D being the middle point of BC; to deter- mine the magnitude and direction of the pressure on the fulcrum, the rod being inclined to the horizon at an angle of 30°, and being without weight. Let BD = CD = a ; and let X, Y, represent the vertical and horizontal pressure exerted by the rod on the fulcrum ; then X = IW, F= - TF; 2 > 2 and, if denote the inclination of the resultant pressure to the vertical, and R its magnitude, B = W, ^ = i7r. (1 7) One end of a beam is connected with a horizontal plane by a hinge about which the beam can revolve freely in a ver- tical plane ; the other end is attached to a weight by means of a string passing over a pulley in the same vertical plane ; to find the position of equilibrium. Let a, b, be the distances of the centre of gravity of the beam from its lower and its higher extremities, W its weight, and d its inclination to the horizon ; let ^ be the inclination of the string to the horizon, and P the weight attached to its extremity ; let / be the distance of the pulley from the horizontal and k from the vertical line through the hinge. Then the position of equilibrium will depend upon the equations P (a + b) sin (0 - 0) = Wa cos 0, (a + h) sin (^ - 0) = ^ sin

+ c. (19) A uniform beam rests with one end upon a given in- clined plane, the other end being suspended by a string from a fixed point above the plane ; to determine the position of equi- librium, the tension of the string, and the pressui'e on the plane. Let 2a be the length of the beam, its inclination to the inclined plane, TV its weight, and M the pressure which it exerts on the inclined plane ; let T be the tension of the string, c its length, and ^ its inchnation to the inclined plane ; also let b be the distance of the fixed point from the plane : and a the incli- nation of the plane to the horizon. Then the position of the beam will depend upon the two equations 2 sin (^ - 0) sin a = cos

= TV (2); and taking moments about A, Wa cos 6 = P .2a sin (

vith its lower extremity on a horizontal, and its higher against a vertical plane ; having given its length, the position of its centre of gravity, and the coefficients of the fi-iction of the horizontal and of the vertical plane, to find its position when in a state bordering on motion. If a, h, be the distances of the centre of gravity of the beam from its lower and liigher extremity ; fx, {.i , the coefficients of friction between the beam and the horizontal, and between EQUILIBRIUM OF A SINGLE BODY. 73 the beam and the vertical plane ; and d the inclination of the beam to the horizon ; then tan 6== ^^ . fx {a + 0) (7) A straight uniform beam is placed upon two rough planes, of which the inclinations to the horizon are a and a, and the coefficients of friction tan X and tan X' ; to find the limiting- value of the angle of inclination of the beam to the horizon at which it will rest, and the relation between the weight of the beam and each of the two perpendicular pressures upon the planes. Let B be the required limiting angle ; M, R , the pressures on the planes ; and W the weight of the beam. Then 2 tan = cot (a' + X') - cot (a + X), R W ^ E cos X sin (a' + X') sin {a-\^■ a + X') cos X' sin (a - X) ' (8) A uniform and straight plank rests with its middle point upon a rough horizontal cylinder, their directions being perpen- dicular to each other ; to find the greatest weight which can be suspended from one end of the plank without its sliding off the cylinder. Let W be the weight of the plank, and P the attached weight ; r the radius of the cylinder ,Z« the length of the plank, tan X the coefficient of friction. Then P will be given by the relation P rX^ W~ a-rX' (9) A uniform beam AB, (fig. 43), of which the end B presses against a rough vertical plane CD, is supported by a fine string A C attached to a fixed point C in the plane ; to find the position of the beam when bordering upon motion. Let the point B be on the point of ascending ; /x = the co- efficient of friction; a = the length of the beam, CA = I, ^ACB = B, ^ = A.ABD. Then B may be found from the equation (4a- - U~ - fjcp) tan' B - 2^1' tan + 4fr - f = ; 74 EQUILIBRIUM OF A SINGLE BODY. and then

; CE is a beam, moveable about a hinge C in the vertical line through D, and pressing against the beam AB from the extremity B of which a weight P is suspended ; to de- termine the positions of the beams for equilibrium, ha\'ing given that CD is equal to ^Z> or BD. Let AD=CD = BD = a, /^ACD=Q', GC=h, G being the centre of gravity of the beam CE; R = the action and reaction of the two beams at A; TF= the weight of the beam CE. Then for the equilibrium of CE, taking moments about C, we have R . 2a co^ d = W .h ?>\n d ; and for the equilibrium of AB, taking moments about D, R.a cos = r.a sin 20, or R=^ 2P sinO; 76 EQUILIBRIUM OF SEVERAL BODIES. from these two equations, by the elimination of R, we get Wh sin 9=2Pa sin 2$= 4Pa sin 9 cos 6, hW and therefore 6=0, or cos 9 = — ^r ; which determine the required positions of the beams. (2) Two sphei'es and O, (fig. 45), rest upon two smooth inclined planes AC and AC, and press against each other; to determine theii* position. Let W, TV, be the weights of the spheres 0, 0'; R their mutual action and reaction ; a, d, the inclinations of the planes AC, AC, to the horizon; 9 the inclmation of the line 00', joining the centres of the spheres, to the horizon. Then for the equilibrium of the sphere O, resolving forces parallel to AC, R cos {a + 9)= TV sin a ; and for the equilibrium of the sphei'e 0', resohdng forces pa- rallel to AC, R cos (a - 9) = TV ' sin a. Eliminating R between these two equations, TV sin a cos (a -9)= TV sin a cos (a + 9), TF tan a (1 + tan a tan 9) = TV tan a' (1 - tan a tan 9), TF' tan a - TV tan a and therefore tan 9 = ( TT^' + TV) tan a tan a ' (3) Three spheres 0, 0', , (fig. 46), are placed in contact within a hollow sphere ; a vertical plane thi'ough the centre of the hollow sphere being supposed to contain the centres of the thi'ee solid spheres ; to find theii* positions of equilibrium. Let Cbe the centre of the hollow sphere; 0, O, 0' , the centres of the solid spheres ; join OC, O'C, O C; let TF, TV, TV , be the weights of the thi-ee spheres ; CO = r, CO' = r , CO = r ; l-OCO'=a, 1.0' CO = a; = the inclination of OC to the horizon. Then since the actions of the hollow sphere on the solid ones all three pass through the point C, we have for the equilibrium of the solid spheres, taking moments about C, and observing EQUILIBRIUM OF SEVERAL BODIES. 77 that if each of the spheres be in equilibrium singly they would likewise be at rest connectedly, JVr cos (9 - a) + W'r' cos + W r cos {B + a) = 0, IVr (cos a + sin a tan 9) + W'r' + W'r (cos a - sin a" tan ^)= ; , ^1 J. ^ f\ W'r" cos a -i-W'r' + Wr cos a and thereiore tan 9 = -t-rr,i-T, — . , — ^^t- — • • w r sm a — Wr sm a (4) A sphere and cone of given weights are placed in contact on two inclined planes, the intersection of Avhich is a horizontal line ; to determine the cii-cumstances of equilibrium. Let W, W, be the weights of the sphere and the cone, which we may suppose to be applied at their centres of gravity G, G', (fig. 47). Let M be the action of the plane AB upon the sjjhere, and S the mutual action of the sphere and cone : if ^ denote the semiangle of the cone, then evidently the line of action of S will make an angle with the plane AB'. The plane AB' will exert at right angles to itself an action upon every element of the base of the cone ; the resultant of all these actions will be some force M' applied at some point E of the base of the cone in the line AB'. Let a, a, be the inclinations of the two planes to the horizon. For the equilibrium of the sphere w^e have, resolving forces parallel to the plane AB, W sin a = S cos {a + a - (p) (1), and resolving forces at right angles to the plane, B = W cos a + S sin {a -^ a - (p) . . . . (2) ; the equation of moments is an identical equation, since all the forces wliich act upon the sphere pass tkrough its centre. Again, for the equilibrium of the cone, resolving the forces which act upon it parallel to the plane AB', W sin a = S cos (p (3) ; resolving forces at right angles to the plane AB', R' = W cos a + S sin (j, (4), and taking moments about G', the lines BH, mG', being represented by x, y, B!x = Sy cos (5). 78 EQUILIBRIUM OF SEVERAL BODIES. From the equations (1) and (3), TV sin a cos (a + u - (b) (6), JV sin a cos from which tan ^ may be readily detennined : this relation is the only condition to -which the cone and sphere are subject to secure equilibrium ; as "will be evident when it is observed that the three equations (2), (4), (o), introduce fom- unknown quantities H, B! , x, tj, each of the three equations at least one, which are not involved in (1) and (3). Froin this it is evident that there will be an infinite number of positions of equilibrium, or that if only have the value given by (6), the cone and sphere will rest in contact in whatever manner they may be placed on the two planes, and whatever be their magnitudes. The values of ^ being determined by (6), S will be deter- mined by (1) or (3), and therefore R, R, from (2), (4), respec- tively. Then from the equation (5) we may determine x, pro\'ided that i/ be given; and y can be given only by our knowing the magnitudes of the cone and sphere, and the particular position of equilibrium in which we may choose to place them. (5) Two uniform rods AC, A'C, of which the lower extremi- ties are situated in the same horizontal plane, and prevented from shding, lean against each other at the point C, and are in equdibrium ; to determine the relation between their angles of inclination to the horizon, the small area of mutual contact at C being vertical. Let W, W, be the weights of the rods A C, A' C, respectively, and (}t, (f>', theu- angles of inclination to the horizon ; then W tan = TV tan (2); EQUILIBRIUM OF SEVERAL BODIES. 83 and taking moments about 0, jiRr + 2aY = Wa cos (3). Again, for the equilibrium of the wheel O , we have, taking moments about the point of contact of this wheel with the plane, X'r - Pr sin ^, or X' = P sin ^ . . . . (4). From the equations (1) and (4), observing that X' is by the nature of action and reaction equal to X, we get /«i2 = (2P + TT) sin (5). Again, from (2) and (3), firR + 2a (P + W) cos - 2aR = Wa cos ^, (2a - ixr) R = a (2P +W) cos ^ (6). From (5) and (6) we obtain for the requii-ed inclination of the plane, tan ^ = 2a - fxr Cor. Having ascertained 0, we know It from (5) and X' or X from (4), and therefore Y from (2) ; also F being the only force actmg on the wheel 0' which does not pass through its centre, it is evident that F must be equal to zero. (4) Two equal beams AC, BC, are connected by a smooth hinge at C, and are placed in a vertical plane with theh lower extremities A and B resting on a rough horizontal plane ; from observing the greatest value of the angle A CB for which equilibrium is possible, to determine the coefficient of friction at the ends A and B. If j3 be the greatest value of Z. ACS, and /x be the coeffi- cient of friction at each of the ends ; then H = 1 tan |j3. Sect. 3. Systems of Beams. (1) At the middle points of the sides of any polygon ABODE . . . . (fig. 52), and at right angles to them, are ap- plied a series of forces P, Q, R, . . . ., respectively proportional to the sides ; the sides of the polygon are perfectly rigid, and g2 84 EQUILIBRIUM OF SEVERAL BODIES. capable of moving freely about the angular points A,B, C,D,...; to determine the form of the polygon that it may be in equi- librium, the lengths of the sides being given. Let p, q, r, s, . . . . denote the mutual actions of the sides of the polygon at the angles A, B, C, D, . .. ., of which the directions -will lie in certain straight lines bB^, cCy, dDB, .... For the equilibrium of the side BC we have, resolving forces at right angles to it, Q = ^ sin CB(3 + r sin BCc (1); resolving forces parallel to BC, q cos C5/3 = r cos BCc (2); and taking moments about the middle point of BC, q sin CB^ = r sin BCc (3). Dividing (3) by (2), we have tan CB^ = tan BCc, and therefore A CB^ = ^BCc (4) ; hence also, from (2) or (3), q -= r (5). Again, from (1) and (3), we have Q= 2r sin l^BCc; in precisely the same manner we may find that R= 2r sin L.DCy, J ^, p Q sin Z BCc and tnererore — - = r— r- : R sin ^DCy' but by hypothesis Q BC ^ sin /.BDC B" nC^ sin Z.CBB'' h sin ^B Cc _ sin aB DC ^^rZ~DCy ~ sin z. CBB ' but from the geometry it is evident that L.BCC+ jLDCy^ lBDC^ L.CBB; hence we readily see that /^BCc= l-BBC (6). EQUILIBRIUM OF SEVERAL BODIES. 85 In just the same "way we might proye that and therefore by (4) z.JBDC= ^BAC (7). From this relation (7) it is plain that a cdrcle passing through the tliree points A, B, C, must pass likewise through the point D; similarly we might shew that this circle, since it passes through B, C, D, must likewise pass through E, and so on indefinitely ; hence we see that when the sides of the polygon are arranged consistently with equilibrium, all its angular points must be situated in the cu'cumference of a single circle. From (5) we gather that p = q = r = s = ...., or that the mutual pressures at all the angular points are equal. It is evident also from the relation (6), that all the Hues aa, bft, cy, do, .... are tangents to the ciixle passing through A,B,C,D,.... The value of the mutual pressure at each of the angular points is easily obtained : thus, as we have shewn, Q= 2r sin /-BCc; but since /LBCc is equal to half the angle subtended by BC at the centre of the circle circumscribing the polygon, it is clear that ^BC sin L.BCc hence r = radius x radius Q_ BC and therefore p = q = r = s....= kp, where p denotes the radius and k the ratio between any one of the forces and the corresponding side of the polygon. Fuss ; Memoires de St. Petersh. 1817, 1818, p. 46. The following is a different solution of the same problem : — Let the forces P, Q, R, .... he represented in magnitude by the lines 2AB, 2BC, 2CD, . .. ., to which they are propor- tional. Instead of the force 2AB acting at the middle point of the side AB, apply two forces, each equal to AB, one at the 86 EQUILIBRIL'M OF SEVERAL BODIES. end A and the other at the end B of the side AB ; each of these forces being at right angles to the side AB. Again, instead of the force 2BC acting at the middle point of BC, apply a force BC at C, and a force BC at the extremity B of the side AB, (which we are at liberty to do, because the point B of AB is rigidly attached to the point B of BC,) each of these forces being at right angles to BC. Now, according to this distribution of the forces, the only force which coidd twist BC about C, is the action of the rod AB upon the end B of BC; and therefore for the equilibrium of BC it is necessary that this action shoidd take place exactly along B C. Hence " conversely the action of CB upon BA will take place entirely in the direction CB. Let this action be denoted by B. Thus, the line AB is acted upon at the point 5 by a force AB at right angles to AB, a force BC at right angles to BC, and a force B, in the dii'ection CB: but, by the principle of the parallelogram offerees, the forces AB and BC at B are equiva- lent to a single force AC acting at right angles to AC; hence for the equilibrium of AB we have, taking moments about A, B.AB. sin ^ ABC =^"AC.AB, cos ^BAC, . or B sin l.ABC= AC cos /-BAC. Similarly for the equihbrium of the side CD, R sin /.BCD = BD cos Z.BDC; , ,, c ■ sin Z.ABC AC cos /lBAC and theretore -. ^777^ = -^r^ ^^tTv • sm lBCD BD cos lBDC -But by the geometry, BC . . j^^ sin ABAC _ AC^'^ ^^^ BD sin A ABC . sin Z.BDC BC . „^^ ^C sin ^^Ci>" BD ^^" ^^^^ Hence from these two relations we have sin ABAC cos ABAC sin /.BDC~ cos aBDC tan ABAC =tdn A BDC, aBAC= Z.BDC; EOUILIBRIUM OF SEVEKAL BODIES. 87 which shews, as in the former solution, that the sides of the polygon must be so arranged that its angular points may all lie in the circumference of a single circle. (2) A quadrilateral ABCD, (fig. 53), consists of four rigid rods, which are capable of free motion about the angular points A, JB, C,D; supposing the points A, C, and B, D, to be attached together by strings AC and BD in given states of tension, to determine the geometrical conditions necessary for the equilibrium of the quadrilateral. Let P, Q, represent the tensions of the strings AC, BD. Let K, L, M, N, denote the actions and reactions between the four pairs of points (A, B), (B, C), {C, D), (D, A). The force P acting upon the point A in the direction A C, is equivalent to a force, in the direction AB, _ p sin CAB _ sin ABB BO _ pOB.AB , sin BAB " sin BAB ' A0~ BB.OA' and to some force {F suppose) in AB. Similarly, the force Q acting upon the point B in the direc- tion BBf is equivalent to a force, in BA, = Q ' „ , and some force (G suppose) in BC. Hence clearly the point A is solicited by a force F- N in AB, and a force ^OB.AB ^. .„ ... ^ BB70A- ^ '"^ "^^ ^'^' and therefore for" its equilibrium we have Similarly for the equilibrium of the point B there is G-L^Q, and ^ ^^T"^ " ^^ ^ * " * ' ^^^' From (1) and (2) we have OIKAB _ OC.AB ^ BB. OA'^AC.OB' 88 EQUILIBRIUM OF SEVERAL BODIES. and therefore P . OD Q.OC BD.OA AC. OB' which is the condition for the equilibrium of the quadrilateral. Euler; Act. Acad. Petrop. 1779, P. ii. p. 106. The following is a diiferent solution of the same problem : For the equilibrium of the rod AB there is, taking moments about B, N. BD . sin i.BDA = P . BO . sin Z.BOC; and for the equilibrium of the rod CD, taking moments about C, N. CA . sin L. CAD = Q . CO . sin ^BOC; hence obviously BD sin A ODA BD . AO P . BO or - — = CA sin z. OAD ' AC . DO Q . CO' (3) Four rigid rods AB, BC, CD, DA, (fig. 54), are so joined together that they are capable of revolving freely about the angular points of the quadi'ilateral which they form ; these rods are attached together, two and two, viz. those which are contiguous, by strings aa, 5/3, cy, dS, in given states of tension ; to determine the form of the quadrilateral which shall corres- pond to the equilibrium of the rods. Let A, B, C, D, denote the tensions of the strings aa, b(3, cy, dS. Then the force A in aa upon the point a is equivalent to a force, in BA, 7-. sm aDa . , y-, _ . sm aaD aa j Aa . Da sin AaD "7"^ DA «« . DA ' sin ADa . Aa and to a force, in aD, sin aAa Aa sin Aaa _ > ' ' aa . ^a . Da sin AaD AD " «« • -^^ sin a AD . — - aD = A' suppose. EQUILIBRIUM OF SEVERAL BODIES. 89 But the force A' in aD is equivalent to ^ sin aBD . --— a force m AJD, = A —. ^ = A and to a force in BD, = A sin ABB . ,^^ BA sm ABJJ . -^fT-T DA _ ., Ba . DA _ Aa . Ba ^ Da.BA^ aa.BA' I sin ADa sin ADB ■ J.. Aa = A' ^"' • Da ^ ^, Aa.BD ^ ^ Aa . Aa . BD . -r, , „ AB ~ AB .Da~ aa . DA . BA ' sm DAB . DB Thus we see that the force A, acting upon the point a in the direction aa, is equivalent to the tlii'ee forces . aD . Aa . T, , J J Aa . aB . , t-, j A — -=- m BA upon A, A -—=, m AD upon A, AD .aa AB . aa -, . aA . Aa . BD . 7, 7^ 73 and A — -jr- — -^ m BD upon B. AB . AD . aa Similarly, the force A acting upon the point a in the direction aa, is equivalent to . aB . Aa . -^ . . . Aa . aD . . -^ a A — r^r m DA upon A, A — -^r m AB upon A, AB . aa r ' ^2> . aa , . aA . Aa . DB . 7-, T? r» and A -—^ rvi m DB upon D. AD . AB . aa Now these three forces are equal and opposite to the three former, and therefore the string aa wdth a tension A produces the same effect, and may therefore be replaced by a string BD with a tension . aA . Aa . BD AB .AD. aa ' In the same way we may shew, that the tension of cy is equi- valent to a string BD, of which the tension is equal to cC. Cy.BD CB .CD .cy' 90 EQUILIBRIUM OF SEVERAL BODIES. Hence the tensions of aa, cy, together, are equivalent to a string BD with a tension Aa .Aa.BD Cc.Cy .BD BA .DA.aa^ BC. DC .cy' Similarly it may be shewn, that the tensions i/3, c/S, are equivalent to a sti'ing A C with a tension Bb .B(3.CA D^.Dd .AC AB . CB . b(3 ^ AD . CD . dS Hence, by the result of the preceduig problem, the condition of equilibrium is expressed by the relation OB.OD f B.Bh. Bi3 D . PS . Dd \ BD' \AB . CB . b(5 ^ AD . CD . dSj ■Cy \ C.cyJ' AC \BA.DA.aa^ BC.DC Euler; Act. Acad. Pctrop. 1779, P. 2, p. 106. (4) Three uniform beams AB, BC, CD, of the same thick- ness, and of lengths /, 2/, /, respectively, are connected by hinges at B and C, and rest on a perfectly smooth sphere, the radius of which is equal to 2/, so that the middle point oi BC and the extremities of AB, CD, are in contact with the sphere ; to compare the pressui-e at the middle point of BC, and the pressures at A and D, with the weight of the tlu-ee beams. Let W be the weight of the three beams taken together ; R the pressui-e at each of the points A and D ; and R' the pressure at the middle point of BC. Then R _ Z R' _ 91 XF~4o' Tr'Too' (5) Fom- equal uniform beams AB, BC, CD, DE, (fig. 55), connected together by jomts at their extremities, rest in equi- librium in a vertical plane ; the distances AE and CF, of which the latter is perpendicular to AE and vertical, are given; to determine the conditions of equilibrium. EQUILIBRIUM OF SEVERAL BODIES. 91 If a, (5, be the inclinations of AB and ED, BC and DC, to tlie horizon ; we must have tan a = 3 tan j3. Draw BKoX right angles to AE; let CF = a, AF=h, FK=x, BK= y; then from the equation in a and j3, and the geometry of the figure, we may get a- + 2V - U + a^W + h'j 2a' + 5' - (a* + c^h"" + hj X =-- ^^—5 ^ , V = ^^ — . 2b ^ 2a These values of x and y are obtained by Couplet in his Recherches sur la Construction des Comhles de Cliarpente, in the Memoires de VAcademie des Sciences de Paris, 1731, p. 69. (6) A and C (fig. 56) in the same vertical line are fixed points, about which beams AB, CD, are freely moveable by hinge joints; AB is supported in a horizontal position by CD, with which it is connected by a hinge joint at D, and has a weight suspended at ^ : to find the pressure at C, the weights of the beams being neglected. Let H and V be the horizontal and vertical pressures at C, and P the weight suspended from B, Then and therefore the whole pressure at C is equal to p.abJ'^ ' AC AD\ (7) Two equal uniform beams AB, AC, moveable about a hinge at A, are placed upon the convex circumference of a cu'cle in a vertical plane ; to find theii- inchnation to each other when they are in their position of equilibrium. Let 2a = the length of each beam, 20 = their inclination to each other, and r = the radius of the circle. Then B will be determined by the equation r cos B = a sin^ B. ( 92 ) CHAPTER V. EQUILIBRIUM OF FLEXIBLE STRIXGS. The form of equilibrium assumed by a uniform flexible string sustained at its two extremities and acted on by gra\-ity, at- tracted the attention of Galileo^, who, from a want of sufficient examination, concluded it to be a parabola ; this mistake may have arisen from the fact, that in the immediate neighbourhood of its lowest point it approximates very nearly to the parabolic form. The inaccuracy of Galileo's conclusion was experi- mentally ascertained by Joachim Jungius". This subject having been at last successfully investigated by James Bernoulli^, he proposed the problem of the chainette, the name which he gave to the requu-ed curve, as a trial of skill to the mathematicians of the day. The four mathematicians who succeeded in arriving at correct solutions of the problem were, James Bernoulli, by whom it had been proposed, his brother John, Leibnitz, and Huyghens : their fom* solutions appeared without analysis in the Acta Eruditoriim for the year 1691, Jun. pp. 273 — 282 A demonstration of the results of these four illustrious mathema- ticians was first pubhshed by David Gregory, in the Philosophical Transactions for the year 1697. The form of equilibrium of the chainette or catenary, of which the thickness is supposed to be uniform, ha^dng been thoroughly discussed, James Bernoulli* next dii-ected his atten- tion to more complicated problems of the same character ; he ' Mechanica ; Dialogo 2, p. 13L Geometria Empyricu. ' Acta Eruditorum, Lips. 1690, Mai. p. 217 ; Opera, torn. i. p. 424. * Ada Eruditorum, Lips. 1691, Jun. p. 289; Opera, torn. i. p. 449. EQUILIBRIUM OF FLEXIBLE STRINGS. 93 investigated the form of equilibrium when the thickness varies from point to point according to any assigned law, and, con- versely, determined the law of its variation that the string may hang in assigned curves : he likewise considered the problem of the catenary when the string is extensible, the extension of each element being assumed according to the law established experimentally by Hooke' to vary as the tension. The analysis of these problems, of which the solutions only were published by James Bernoulli, was supplied by John Bernoulli^. The consi- deration of the general conditions of the equilibrium of flexible strings was first attempted by Hermann^, whose investigations, however, were not free from error ; a more accurate analysis was furnished by John Bernoulli*, who has particularly ex- amined various cases of the equilibrium of strings acted on by central forces. Among the numerous mathematicians who afterwards dis- cussed the theory of the equilibrium of flexible strings, may be mentioned Euler^ Clairaut*', Krafft^ Legendre^ Fuss®, Ventu- roir", and Poisson^^ Sect. 1. Free Inextensible String ; general Conditions of Equilibrium. To investigate the conditions for the equilibrium of an inextensible string, of which the density and thickness vary from point to point according to any assigned law ; the accele- rating forces which act upon the string being any whatever. ' De Poientia Restitutivn, or Spring. ' Lectiones MathematiccB in usum HospitaUi , Opera, torn. iv. p. 387. ' Phoronomia, lib. i. cap. 3, and Append. §. v. * Opera, torn. iv. p. 234. * Comment, Petrop. torn, ill.; Nov. Comment. Petrop. torn. xv. and torn. xx. ° MisceUunea BeroUnensia, torn. vir. p. 270, 1743. ' Nov. Comment. Petrop. torn. v. p. 143; 1754 and 1755. " Mem. Acad. Par. 1786, p. 20. * Nova Acta Petrop. torn. xii. p. 145, 1794. '" Elements of Mechanics, by Cresswell, Part i. p. 62. " Traits de Mecanique, torn. I. p. 564, seconde edition. 94 EQUTLTBRIUM OF FLEXIBLE STRINGS. Let APB (fig. 57) be any portion of the string in a position of rest ; Pp being a small element of its length ; z, xj, z, and X + ^x, y + hj, z + "^z, the co-ordinates of P and p respectively ; s the length of the string reckoned from some assigned point up to P, and 5 + ^5 the length up to ^^ ; t the tension of the string at P. The resolved parts, parallel to the axes of x, y, z, of the force exerted upon the point P of the element Pp by the portion AP of the string, will evidently be -t — , -t^ -t~; els ' ds ' ds ' and therefore, since each of these tliree forces must be some function of s, it is plain by Taylor's theorem that the resolved parts of the force exerted on the element Pp by the portion qB of the string, will be ^dx d ( ^ dx\ ^ ds ds V ds t$ + 4.(tf)ss, ds ds \ ds t'!;,i(t"S\h. ds ds \ ds Again, let X, Y, Z, be the sums of the resolved parts of the accelerating forces which act upon the element Pp); p the density of the string at P, and k the area of a section at right angles to its length at that point. Then clearly the mass of the portion Pp of the string will be kp^s, which therefore for a constant value of ^s will vary as kp; hence e%-idently the product kp, which we will call m, may be taken to measm-e the 7?iasstveness of the string at the point P ; it will be con- venient to call it the unit of mass at the point P. The impressed moving force then of the element Pp, vrill have for its resolved parts parallel to the co-ordinate axes, mXh, f)iYBs, mZSs. Hence clearly for the equilibrirmi of Pp we must have, equating to zero the sum of the resolved forces which act upon EQUILIBRIUM OF FLEXIBLE STRINGS. 95 it parallel to each of the three axes, and dividing the three resulting equations by Ss, as \ as d_ ds d_ ds ds ds mZ 0, Oj («); which three equations constitute the conditions of equilibrium of the entire string. By the elimination of t we readily obtain the three following equations, dx fmYds = dy fmXds, dy jniZds = dz fniYds, dz fmXds = dx fmZds ; any two of which will be differential equations to the required curve of equilibrium. Cor. 1. From the equations (a) we have also t—-=^- fmXds, t ~ = - [m Yds, t -^ = - fmZds ; ds •' ds -^ ds ■' squaring and adding these equations, and observing that = 1 (h), dx'' dif dz^ ds"' ds" ds^ we obtain for the value of the tension at any point, f = {JmXdsJ + {Jni YdsJ + (fmZdsJ. We may obtain also another expression for the tension : differentiating (b) with respect to s, we get dx d^x dy d^y dz d'^z ds ds^ ds ds^ ds ds^ • • • • V ; » hence, multiplying the three equations (a) by dx, dy, dz, in order, and adding the resulting equations, we have, by the aid of (h) and (c), t = C - Jni (Xdx 4 Ydy + Zdz), where C is an arbitrary constant. 96 EQUILIBRIUM OF FLEXIBLE STRINGS. Cor. 2. If the whole string lie entirely within one plane, let the plane of xy be so chosen as to coincide Tvith this plane ; then the three differential equations to the string will be reduced to the single one dx fm Yds = dy JmXds {d) ; and the two formulae for the tension will become f = (fmXdsf + {fm Ydsf, t = C-fm(Xdx + Ydy). These two formulae for the tension, and also the differential equation (d) to the string, coincide with those given by Fuss, Memoires de St. Petershourg, 1794, p. 150, 151. Sect. 2. Parallel Forces. (1) A flexible string fixed at any two points A and B, (fig. 58), is acted on by gra^dty; supposing the unit of mass to vary according to any assigned law as we pass from one point to another, to find the equation to the catenary of rest ; and conversely, the curve being known, to' determine the law of the xinit of mass. Let the axis of y extend vertically upwards, and let the axis of x be horizontal, the plane xOy coinciding with the plane which contains the catenary. Then since X=0, Y=-g, we have, by the first two of the equations (a) of section (1), d /. dx\ , . ^^=0 (a), ds \ ds Integrating the eqviation {a), we get t'^ -C * ds -^' ds \ ds "-^t"l).n.g (6). -f EQUILIBRIUM OF FLEXIBLE STRIiNGS. 97 ■where C is a constant quantity : let t denote the tension at the lowest point of the curve, then evidently t = C, and therefore dx . . as From {li) and (c), Ave have d dy ds dx and therefore r ~ = \ ma ds ; dx J dv but evidently at the lowest point of the catenary -t- = ^^ ^^i' which gives the tension at any point of the catenary when its form is known. John Bernoulli ; Lectiones Matheinaticce, Lect. 38, 39, 40; Opera, tom. iii. (2) A flexible string AOB, (fig. 60), fixed at two points A and B, is acted on by gravity ; the unit of mass at any point P varies inversely as the square root of the length OP measured from the lowest point O ; to find the equation to the catenary. H 98 EQUILIBRIUM OF FLEXIBLE STRINGS. Let the origin of co-ordinates be taken at 0, x being horizon- tal, and y vertical, and the plane of xy coinciding with the plane of the catenary ; also let be the origin of s. Then if fx be the unit of mass at a length c from the lowest point, j^ m = fx-, and therefore by (1, cT), a being in the present case zero, we have ax Jo i S hence, putting for the sake of brevity r i' (5' we get fJy _ ( s\ dy^ _ s r, d dif _ ds I dx da? dx \ .2\1 d dtf _ ds ( ^ dy~ ' d?' ' n.d_ d£ dx dx' 1; integrating with respect to x we obtain but x= 0, -^ = 0, simultaneously ; hence C = 2(5, and therefore ^''O-ih-^^'3 W; squaring and transposing, 4/3^ g ==(:. + 2/3/ -4)3% 2(i dy = {(x + 2j3)- - 4/3'}' dx; EQUILIBBirM OF FLEXIBLE STRINGS. 99 integrating we have C+ 2/3?/ = K^ + 2/3) (a;'- + 4l5xf - 2/3' log {a: + 2/3 + {z" + 4(Bxf} ; but X = 0, y = 0, simultaneously ; hence C=-2/3Mog(2/3); hence^ eliminating C, 2/3y = Kx + 2/3) (:r' + 4/3^)' - 2/3' log "^ ^ ^^ ""^^g' ^ ^^'"^ . which is the required equation to the catenary. Cor. From (a) we get ds X + 2^ dx^~2^ ' and therefore by (l,f), ds T , ^^- which gives the tension at any point of the curve. John Bernoulli; Led. Math., Opera, tom. in. p. 497. (3) To find the law of variation of the unit of mass of a catenary acted on by gravity that it may hang in the form of a semicu'cle Avith its diameter horizontal. The notation remaining the same as in (2), the equation to the catenary will be X- = lay - If, where a denotes the radius of the semicircle : hence d - 01? = {a - yj, y = a -{a^ - af)^ ; dy _ X dry a' ax ^ 2 2\2 ttX / 2 2\5 (a - x) [a - X) , di' , dy^ a^ ds a also — ^=1+^- dx' dx~ a^ - o(? dx . , ■> i (rr - x-y and therefore by (1, e) d^ dx^ m fj db g a ~x g{ci-yf dx or the unit of mass at any point varies inversely as the square of its depth below the horizontal diameter of the semicircle. H 2 100 EQUILIBRIUM OF FLEXIBLE STRINGS. Cor. By {'^,f) we have for the tension at any point ds Ttt ra clx . „ „ 1 a- y John Bernoulli; Opera, torn. iii. p. 502. (4) To find the length of a uniform chain ALB, (fig. 61), suspended from two points A and B in the same horizontal line, when the stress on each point of support is equal to the whole weight of the chain ; to find also the depth of the lowest point L of the chain below the line AB, and the direction of its tangent at A or B. Let yCLO be vertical, OL being equal to a length of the chain of which the weight is eqiial to the tension of the lowest point L, Ox horizontal ; PM at right angles to Ox. 031= x, PM=y, OL = c, ALB^l, AC=BC^a. Then the equation to the curve will be X X tj^lcie^'+e^'") (1), a a and also / = c (e'' - £ ") (2). Let m denote the unit of mass of the chain, which will be the same at all its points ; then the tension at P will be equal to X X mgy = 5 meg (e'' + f '^), a a and therefore at -B to \ meg {f + £ ") j but by the hypothesis the tension at B is equal to mgl, and therefore by (2) to a a. meg if - £ J a a a u hence g meg {(P + £*')= meg (/^ -£*'); 1 ,c _ 3 2 1 — 2 ,r=3, - = log, 3, -=llog. 3 (3). c c EQUILIBRIUM OF FLEXIBLE STRINGS. 101 Hence from (2) we have 2a /„i 1\ 4a 1 = ^ ^ 3-/ 3'losr 3 0£ whicli gives the length of the chain. Again, putting x - a, we have from (l), a a a a and therefore CZ = (i£'^ + i£ ^^-1)0 3U3-' ,^ 2« ^f^„^(3)^ 2 y log, 3 J'-^ 2a 2a g, y log, 3 log, 3 g4 which gives the depth of the lowest point of the chain below the Hne AB. Again, from (1) we have and therefore, denoting the inclination of the chain at B to the horizon, 1 \ 1 tan^^K^''-^ '^^n^'~" V r' 3' hence 6 = - . ^ 6 (5) A uniform string A'ALBB' (fig. 62) is placed over t^A'o supports A and B in the same horizontal line, so as to remain in equilibrium ; ha\dng given the length of the string, and the distance of the points of support, to find the pressure which they have to bear. Let L be the lowest jDoint of the curve ALB, OLy a vertical line through L, where OL is equal to a length of the cham, the weight of which is equal to the tension at L ; Ox horizontal. 102 EQiriLTBRIUM OF FLEXIBLE STRINGS. Then Ox, Oy, being taken as the axes of co-ordinates, we shall have for the equation to the curve ALB, putting OL = c, X X y = \c{? + ,~~^) (1); and if m be the unit of mass at each point of the string, the tension at P will be equal to X X mgy or I meg (e'^ + £ '^) ; hence, if -4C= BC = a, the tension at B will be equal to a a lmcg{/+ r^ (2). But the tension at B is evidently equal to the weight of BB , and therefore, if BB = s, to the expression mgs ; hence a a mgs = \ meg (t'' + £ % a a or s = i c (e^ + £~ •') (3). Suppose that the length of the whole string AALBB is 2l : then the length of the portion LBB will be I, and l-s wiU be the length of BL. Hence, by the nature of the catenary, a a l-s = \c{^ - f^) (4). Adding together the equations (3) and (4), we obtain a whence c is made to depend upon the known quantities a and I: hence the expression (2) for the tension at B is known. Differentiating (1), we get but, if ZP = s, 5 _•! els' -^ _f dy £*' — £ '^ hence evidently -/-, = , ^ ds 5 -I' EQUILIBRIUM OF FLEXIBLE STRINGS. 103 and therefore if denote the angle between the line BB' and the curve BL at B, a a COS ^ = V~T? " ' ^^^' t' + E ' Let P denote the pressure on the point B, and t the tension of the string at B; then P^ = 2r- - 2t' cos <^ = 2r^ (1 - COS 0); and therefore from (2) and (5), a a £ + £ J a a a 2a = wV/ (1 + e~~) 2a X P = meg (1 + £ " ) , which gives the requii'ed value of the pressure, c having been previously determined. (6) A uniform chain ABC (fig. 63) is suspended from a point A above an inclined plane BS; having given the angle which the chain at the point of suspension and which the plane makes with the horizon, and also the length of the whole chain, to find the length of the portion BC which is in contact with the plane. Let ABLA' denote the catenary, of which AB is an arc, L being the lowest point. Let P be any point in the curve AL ; the inclination of the curve at P to the horizon, t the tension at ^ ; a, /3, the values of at A, B, respectively ; c the length of chain of which the weight is equ 1 to the tension at L; m the unit of mass of the chain; LP = s, ABC=l, BC=r. Then, by the nature of the catenary, t cos j3 = mcff (1), s = c tan ^ (2). 104 EQUILIBRIUM OF FLEXIBLE STRINGS. Xow it is e^-ident that the tension at B is equal to mgl' sin /3; hence from (1), mgT sin /3 cos j3 = meg, c = /' sin /3 cos /3 . . . . (3). Again, from (2) we have LBA = c tan a, LB = c tan /3, and therefore I - l' =. c (tan a - tan /3) ; hence, fi'om (3), I - r = r sin j3 cos j3 (tan a - tan j3), / cos a = r (cos a + sin a sin /3 cos j3 - sin* /3 cos a) = /' cos j3 . cos (a - /3) / cos a /' cos )3 COS (a - /3) (7) AOB (fig. 60) is a flexible string acted on by gra^-ity, and is in a position of rest ; the unit of mass at any point varies as the cosine of the angle at which an element of the curve at the point is inclined to the horizon ; to find the equation to the catenary. Assuming m = j3 -r- , where /3 is some constant quantity, the equation to the catenary will be o 2- ^9 which shows that the catenary is the common parabola. James Bernoulli; Act. Erudit. Lips. Jun.; Opera, torn. i. p. 449. John Bernoulli ; O/J^ra, tom. iii. p. 501. (8) To find the equation to the catenary when the unit of mass varies as x cos ^, where ^ is the angle of incUnation of the element of the curve at any point to the horizon. Assuming m = ^x -j- , the requued equation will be which belongs to a cubical parabola. James Bernoulli ; lb. John Bernoidli ; lb. EQUILIBRIUM OF FLEXIBLE STRINGS. 105 (9) To fiud the equation to the catenary when the unit of mass varies as x^ cos ^. i (Ix Assumin» 7n = Bx'^ — r- , the equation will be as 16/j3V= 225 ry. James Bernoulli ; lb. John Bernoulli ; lb. (10) To find the equation to the catenary when the unit of mass varies as y" sin (^, where n is any positive quantity. If the origin of co-ordinates be so chosen that the axis of x passes through the lowest point of the catenary, and that y - oo when x= 0, the required equ.ation will be (n + 1) 7- W no 0^ ' James Bernoulli ; lb. John Bernoulli ; lb. (11) To find the law of the variation of the unit of mass when the catenary is the common parabola. The construction and notation being the same as in (2), 2r ff (a^ + 4x-y a being the latus rectum of the parabola. John Bernoulli; Opera, tom. iii. p. 504. (12) A chain suspended at its extremities from two tacks in the same horizontal line, forms itself into a cycloid ; to find the unit of mass at any point of the string and the weight of the arc between this and the lowest point. Let 10 denote the weight of the arc ; then taking the ordinary equations to the cycloid c-r = « (0 + sin 9), y = a (I - cos 6), we shall have .. ^ (sec I ey Aag ,ic = T tan 2 0. (13) One end of a heavy chain is attached to a fixed point A, and the other to a weight which is placed on a rough horizontal 106 EQUILIBRILM OF FLEXIBLE STRINGS. plane passing tlirough A, and the chain hangs through a slit in the horizontal plane ; to find the greatest distance of the weight from A, at wliich equilibrium is possible. If a be the length of the chain, z the greatest distance of the weight from A, fx the coefficient of friction, and n twice the ratio between the given weight and that of the chain, (14) A uniform chain is suspended from two tacks in the same horizontal line at a distance 2a from each other; to determine the length of the chain that the stress on the tacks may be a minimum. Let c denote a length of the chain of which the weight is equal to the tension at the lowest point; and let / denote the requii-ed length of the chain. Then 0; from the former equation - and therefore c is to be determined^ c and then I will be given by the latter. If for instance 2a = 10 feet, then c = 4 . 168 feet nearly, and Z= 12 . 578 feet nearly. Diarian Repository, p. 644. (15) A chain acted on by gravity hangs in the form of a curve, of wliich a^y = .r* is the equation ; to find where the unit of mass is a maximum, and its maximum value. When m is a maximum, x and y being the co-ordinates of the point, a , 2.3*r x = —,y = la, 711- - r ""^ The law of the mass of the chain is erroneously investigated in the Lady's and GevdlemarCs Diary iox the year 1745; see also Diarian Repository, p. 435. EQUILIBRIUM OF FLEXIBLE STRINGS. 107 Sect. 3. Central Forces. (1) To find the equation to a flexible string in a position of equilibrium under the action of any central attractive force. Let APB (fig. 64) be any portion of the string ; S the centre of force ; P any point in the string, PT o. tangent at this point ; SY a jierpendicular from ^S* upon PT; 2^ ^ point of the string indefinitely near to P, and pk a tangent at p. Also let OP, Op, be the normals at P, p, being therefore the centre and OP the radius of curvature atP ; let OP produced iweetpJc in Jc. OP = p, SP = r, SY=p, z.SPT= (j>, Z.Wp = xp, m = the unit of mass at P ; t - the tension at P and t + dt at J3 ; Pp = ds, F = central force at P. Then for the equilibrium of the element Pp we have, resolv- ing the forces which act upon it at right angles to PT, Fmds sin = (^ + dt) cos /j^-O = (^ + df) sin -tp, or retaining infinitesimals of the first order, ds Fmds sin ^ - tip - t — ; P and therefore Fm sin ^ = - («) ; P and resolving forces parallel to PjT we have Fmds cos (p = (t + dt) sui Okp - t = (^ + dC) cos 'ip - t, or, retaining infinitesimals of the first order only, Fmds cos ^ = dt; and therefore, ds cos being equal to dr, Fmdr = dt (b). From the equation (a), since dp p = - T -j2 and sin ^ = di) we have Fmdr + J- t =^ ; P 108 EQUILIBRIUM OF FLEXIBLE STRINGS. and therefore from (b) dp dt P t log (7^0 = log ^'y where C is an arbitrary constant ; and therefore C C ,, ^^=7^7:F>;^' ^'^' which is the equation to the catenary in j^ ^^^ ^ when the form of jP is known. Let be the angle between SP and any fixed Hne ; then r dO P = ,, (dr + rdOy and therefore from (c), putting jFjyidr = R, Bi^ dd = C(dr + r dOJ, RYde'= C\dr' + rdeP), and therefore de= ^''' (d), r(RY- Cr-f the differential equation to the catenary between r and 9. This is the form in which the solution is given by John Bernoidh.' The value of the tension at any point of the catenary is given by (b), when the expression for F in terms of ?• is knoAvn. The relations at which we have arrived may be deduced from the general equations of equihbrium of section (1); the method however of the tangential and normal resolution is more con- venient in the case of central forces. If the central force be repulsive instead of atti-active, we must replace i^by - i^, wherever it occurs in the above formulcC. (2). To find the form of the catenary when the central force is attractive and varies inversely as the square of the distance ; the unit of mass being invariable. ' Opera, toin. iv. p. 238. EQUILIBRIUM OF FLEXIBLE STRINGS. 109 Let A OB (fig. 65) be the catenary; *S' the centre of force; SO the radius vector which meets the curve at right angles. T = the tension at 0, and SO = c. Then if k denote the attraction at the distance c, R = I Fmdr = \ k - mdr = C , J J r r where C is an arbitrary constant: but, by (1, h), t = M, and therefore T = C - mice; hence ' t - M = r + mkc («)• r Hence, from (l, cT), we have Cdr r ; I r -f mkc ^ | r" - C~\ d9 = *' 11 - iv ^ but from (1, c% since p ^ c and ^ = r at the point 0, we see that C = CT ; therefore crdr dH = ^(r^mkc-'!^y-^A^' For the sake of simplicity put r = nmkc ; then ticdr dB r \[ n + \ — )/••'- ifc: IV ncdr r {{n + 1)V^ - 2 (;z + 1) CT + c" - wVj' the equation to the catenary resulting from the integration of this differential equation will be of three different forms accord- ing as 11 is greater than, equal to, or less than unity. First, suppose that n is greater than unity ; then the integral of the equation will be, supposing that = when r = c, {n-\)c 110 EQUILIBRIUM OF FLEXIBLE STRINGS. Secondly, suppose that n = I ; then the equation to the cate- nary will be, if = when /• = c, c Thii'dly, let n be less than unity ; then, if as before 0=0 when r = c, the equation will be ^(^-n^)in+ ^-i^-n')^-n = ^ |^. _ (l _ n)c]. Again from (l , c) we have, since C = ct, and therefore by (a) c- ct nc , jnkc , c T + mkc w + 1 — r r hence putting r = oo we have p = , which shews that the n + \ thi-ee catenaries, corresponding to the three values of 7i, have all of them asymptotes passing within a distance from the centre of force. Put r = in the equations to the tkree curves, and we get for the inclinations of the pair of asymptotes of each to the line SO, n 1 1 ^ n 1 + (1 _ jff cos — , and log ^ . John Bernoulli ; Opera, tom. iv, p. 240. Whewell's Mechanics, 3rd edit. p. 183. (3) To find the equation to a uniform catenary A OB, (fig. 65), acted on by a central force tending to S, the intensity of which varies as the /x'^ power of the distance ; the tension at being (1 + iif" of the weight of a length SO of the string, each element of which length is supposed to be acted on by a constant force equal to that at O and towards S. The notation remaining the same as in (2), the equation to the catenary will be = COS {fi + 2) d. EQUILIBRIUM OF FLEXIBLE STRINGS. Ill (4) To find the equation to a uniform catenary SAOB, (fig. 66), acted on by a central repulsive force emanating from S, at which the two ends of the string are fastened, the intensity of this force varying inversely as the fx^^ power of the distance ; the tension at being (^ - 1)* of the weight of a length SO of the string, each element of which length is supposed to be acted on by a constant force equal to that at and from S. The notation remaining the same as before, the equation to the curve will be = cos (ju - 2) d. Sect. 4. Constrairied Equilihrium. (1) A flexible string ab, (fig. 67), acted on by gravity, rests on the arc of a curve APB in a vertical plane ; to find the tension of the string and the pressure on the curve at any point. Let P, p, be any two points of the curve very near to each other; PO, pO, normals at these points, the point being the centre of curvature when p approaches indefinitely near to P • let ax, ay, be the axes of x, y, the former being horizontal, the latter vertical ; aP = s, Pp = ds ; t = the tension at P and t-{- dt at/) ; R = the unit of pressure on the curve at P, m = the unit of mass of the string, LPOp = ^, PO = p. Then, resolving forces which act on the element Pp of the string, parallel to the tangent at P, we have {t + dt) cos (j) - t = mgds . -j- , or, neglecting infinitesimals of higher orders than the first, dt - mgdy ; integrating and observing that t is equal to zero when y = 0, we get i = nm (1)' which gives the tension at any point of the string. Again, resolving the forces on the element Pp parallel to the normal OP, dx mgds . — + (^ + df) sin ^ = Hds, 112 EQUILIBRIUM OF FLEXIBLE STRIXOS. or, neglecting infinitesimals of orders higher than the first, clx (p „ '' (Is (Is but %- is equal to - ; hence we have for t]\e pressure on the ds p curve at any point _ dx t U = 7ng —- + - ds p (dx y (2) Two equal weights Q, Q, are suspended at the extremi- ties of a flexible string hanging over a smooth curve in a vertical plane ; to find the pressure at any point of the ciu've, the weight of the string being reckoned inconsiderable. Let APB (fig. 68) be the curve; OP, Op, normals at two consecutive points P, p; 6 the inclination of OP to some assign- ed line in the plane of the curve, POp = dd ; PO = p, AP = s, Pp = ds ; 2^ = the unit of pressure on the curve at the point P ; t = the tension of the string at P, t + dt == the tension at p. Then for the equiHbrium of the element Pp of the string we have, resolving forces at right angles to the tangent at P, (t + (If) sin dO = pds =ppdB, and therefore, retaining infinitesimals of the first order, t dO =ppdO, t =2^9 (!)• Again, resolving forces parallel to the tangent at P, (t + dt) cos d9-t=0, and therefore, retaining infinitesimals of the first order, dt=0, t= constant ; but evidently at A the tension is equal to Q ; hence t = Q. Hence from (1) we have Q = pp, p = - • p Cor. The whole pressure on the curve AB is equal to fpds =Qf-= aj'j dd = Q(0, - 6/,), EQUILIBRIUM OF FLEXIBLE STRINGS. llo If the tangents at the points where the string leaves the curve be vertical, we have irQ for the whole pressure along the curve ; if they be not vertical there will of course be pressures at the points A, B, in addition to the pressure along the curve. Euler; Nov. Comment. Petrop. 1775, p. 307. Poisson; Traite de Mecanique, torn. i. ch. 3. (3) To find the pressure on a curve AB, (fig. 69), when two weights Q, B, balance each other over it by means of a string of negligible weight, the friction between the string and the curve being taken into account ; and the weight Q, being considered as much greater than B as is consistent with equiHbrium. Let //, be the coefficient of friction ; the rest of the notation being the same as in the preceding problem, Then the friction on the element Pp will be upds, and will act nearly in the direction of the tangent at P. Hence, resolving forces on the element Pp parallel to PO, we have (^ + df) sin dQ = pds = ppdO ; and therefore in the limit tdO = ppdO, t=pp (1 j ; again, resolving forces parallel to the tangent at P, (t + dt) cos dO ~ t + upds = 0, and therefore in the limit dt + /jpds = 0, and consequently by (1) dt + — ds = ; integrating, we get P iogt = -^f'^^ = -pfde = c-^e....(2); J p hence, the values of ^ at ^ and B being Q and B, log Q = C - ne^, log B = C- fjiO^.... (3), and therefore log ^ = /x (0.^ - Bj, —=£'"' ° ' , B TV which expresses the relation which must subsist between Q and B under the circumstances of the problem. 114 EQUILTBRITM OF FLEXIBLE STRINGS. Also, from (2) and (3), log 1 = ^(9,- 9), f=Re"''^''' (4); hence the whole pressm^e along the ciii've is equal to rds J pels = I — t, from (1), J P = ltd9 = life'''^'d9=C^^-e'''^-''; but when 9 = 9^, it is clear that the pressure along the curve is zero ; hence 0=0 £ , and therefore the whole pressure fr'om 9^ to (9,, is equal to In addition to this pressure along the ciu've there are the pressures at the extremities A and B. Cor. If the curve be a semicircle 9^-9^ = v, and we have Q ... R = ' • Euler ; Nov. Comment. Petrop. 1775, p. 316. Poisson; Traite de Mecanique, torn. i. ch. 3. Sect. 5. Extensible Stnngs. If an extensible string of given length be stretched by any force, it is found by experiment that the extension of the string beyond its natiu-al length is proportional to the force. From this it is easily seen that, if the string be of variable length, the extension will vary as the product of the force and the natural length of the string. Hence if a denote the natural length of the sti-ing, and a' the length under the action of a stretching force P, we shall have a = « (1 + \P), where A is a constant quantity depending upon the quality of the strinof. EQUILIBRIUM OF FLEXIBLE STRINGS. 115 This theory was first announced by Hooke, in the form of an anagram, among a list of inventions at the end of his Descrip- tions of Helioscopes, published in the year 1676. The anagram is ceiiinosssttuii, from which may be extracted the proposition, " ut tensio sic vis." He afterwards published a work entitled De Potentia Restitutiva or Spring, in which the theory was developed at large with experimental illustrations, Hooke's theory forms the basis of a memoii* by Leibnitz, in the Acta Eruditorimi for the year 1684, entitled De^nonstrationes Novce de Resistentia Solidorum. For additional information on the subject the student is referred to s'Gravesande's Element. Physic. lib. I. c. 26. (1) An elastic string AC (fig. 70) is suspended from its ex- tremity A, and has a weight attached to it at a point B ; the natural lengths of AB, BC, being given, to find the length of the string ^Cin its present circumstances. Let m denote the unit of mass of the string in its natural state; a, h, the natural lengths of AB, BC, and a, b' , their lengths under the circumstances of the problem ; c the length of a portion of the natural string, the wxight of which is equal to the weight attached to B; let P be any point in AB, and p very near to it, ^P = x, Pp = dx; t = the tension at P and t + dt at p. Then, since by Hooke's Principle the unit of mass of the ele- ment dx must evidently be diminished in the ratio of 1 + A^ : 1, the weight of Pp will be mgdx \^\t' and therefore for the equilibrium of Pj) t + dt + — ^^ - ^ = 0, \ -^Xt (1 +\t) dt ^ mgdx = ; integrating we get ^ (1 + \\t) + mgx = C ; but it is evident that t = nig (a + b + c), when x = 0, and t = mg (b + c), -svhcn ,r = a ; i2 116 EQUILIBRIUM OF FLEXIBLE STRINGS. hence we obtain (a + b + c){l +1 \mg {a^h + c)] ={h + c) [\ + ] \mg (b + c)] + a', and therefore a = a + I \mg {(a + J + cf - {h + cj] = « + ^ \mg (a* + 2ab + 2ac) = « { 1 -f i Xing (a + 2i + 2c)} (1). Again, if Q be any point in BC, BQ = y, and r = the tension at Q, we shall have, as before, r (1 + 5 Xr) + 7ngy = C; but evidently T = m(/J, when y = 0, and r = 0, when y = b' ; hence we have b' = b{\ +\ Xmgb) (2). Hence from (1) and (2), if /' denote the whole length of the string A C, we find that r = a.Jr b + 1 Xmg {a (a + 2b + 2c) + b"^]. (2) An elastic string, of which the unstretched length is a, is placed on a smooth inclined plane the length of which is also equal to a; to find the length which will hang over the plane, the string being stretched by its own weight. Let ^CX)(fig. 71) be the string hanging from the point A in the inclined plane AC; P any point in AC, and j) ^ point near to P ; ^ = the tension at P, T?itA, and r at C; AP = x, Pp = dx ; m = the unit of mass of the string when unstretched, a = the in- clination o{ AC to the horizon. Then, by virtue of Hooke's Principle, the mass of Pj) will be mdx iTXt' and therefore, t + dt being the tension at p, w^e have for the equilibrium of Pp dt^'^^l^dx^O, \ + Xt {\ -\- \{) dt + mg sin a dx = ; EQUILIBRIUM OF FLEXIBLE STRINGS. 117 integrating we obtain t (1 + I Xt) + mgz sin a = C; hence r being the vakie of t when x = a and T when x = 0, we have r (1 + 5 \t) + mga sin a = T (\ ■{- 1 XT), (r - T) {1 + IX(t + T)] + jnga sin a = (1). Let s be the natural length of CD ; then a - s will be the natui-al length of ^C, hence clearly T = mgs, T = mg \s + (a - s) sin a) ; we have then, by (Ij, # (a - s) [1 + 2 Xmg (25 + (a - s) sin a}] = a, \ Xmg (ci - s) {2s + (a - s) sin a} = s, whence s may be determined by the solution of a quadratic equation. If s' be the actual length of the portion CD of the string, we may shew that s = s (\ +2 Xms), and therefore s and s' are both known. (3) A slightly extensible string Aa (fig. 72) is attached to the upper extremity A of the vertical radius ^0 of a circular arc AB along which it rests ; having given its natural length, to find its length as it rests on the arc. Let P, p, be any two points very near together in the string Aa I draw the lines PO, pO; let AP = s, Pp = ds, Z.AOP = 0, /L POp = d(p, AO = a ; m = the unit of mass of the string when unstretched ; t, t + dt, the tensions at P, p) ; s' , ds' , the lengths of AP, Pp, withou.t stretching. Then, by Hooke's Principle, ds = {\ + Xt)ds' (1). Again, for the equilibrium of the portion Pp of the string, we have, resolving forces parallel to the tangent at P, (t + dt) cos (d(j)) + mgds sin (p = t; and therefore, retaining infinitesimals of the first order, dt + mgds' sin = 0; 118 EQUILIBKIUM OF FLEXIBLE STRINGS. hence by the aid of (I), we have (1 + X^) dt + mg sin (1 - X^) = ad(l> - a\td(f) (3). Now from (2) we get approximately t = mag (cos (j> - cos |3), and therefore, substituting this value of t in the small term of the equation (3), ds' = adcp - tna'Xg (cos ^ - cos /3) f/^ ; integrating we get s' + C = a^ - tna'Xg (sin 0-0 cos /3) ; but when = 0, it is e^-ident that s' = ; hence (7 = 0, and we have s' = a(p - mdrXg (sin 0-0 cos /3) ; let ajS' be equal to the natural length of Aa ; then evidently «j3' = a|3 - ma^Xg (sin |3 - j3 cos j3), j3' = ^ - maXg (sin /3 - j3 cos j3) ; but since /3 = /3' nearly, we may substitute j3' for /3 in the coeffi- cient of the small quantity X ; thus we obtain /3 = j3' + waX<7 (sin /3' - i3' cos /3'), which determines the required length Aa. EQUILIBRIUM OF FLEXIBLE STRINGS. 119 (4) Two weights P, Q, (fig. 73), resting on two smooth in- clined planes CA, CB, are connected by a given elastic string PQ; to find theii* position of equihbrium. Let B be the inclination of QP to the horizon ; a, /3, the inclinations to the horizon of the planes CA, CB ; a = the natural length of the string PQ. Then the position of equili- brium will be defined by the two equations, ^ P cot B - Q cot a r,^ f, \P sin a 1 tan 6 = g y. , PQ = a\l + ^ V . P + Q, [^ cos (a - a)j (5) Two equal weights P, Q, (fig. 74), are connected by an elastic string PQ, of which the horizontal line BC is the natural length ; to find the nature of the curves BP, CQ, on which they will always remain in equilibrium with the string parallel to the horizon ; the plane of the curves being vertical. Bisect BC in A, and draw AM vertical ; let AB = a = AC, AM = X, MP = y = MQ ; then the equation to each of the ciu'ves will be (y - aj = 2\aPz, or BP, CQ, are two semi-parabolas of which B, C, are the vertices. (6) An elastic ring J5C is placed round a vertical cone and descends by its own weight ; to find the position of equilibrium. Let (fig. 75) be the centre of the ring in its position of equilibrium, L OAB = a ; lira = the natiu'al length of the ring, and W = its weight ; then BO=a{\ + ~ IFcotaX which determines the required position. ( 120 ) CHAPTER VI. VIRTUAL VELOCITIES. The Principle of Virtual Velocities consists in the following general proposition : " If any assignable system of bodies or points, solicited each of them by any forces whatever, be in [equilibrium ; and we conceive this system to experience consistently with its geome- trical relations any small arbitrary displacement, by virtue of Avhich each point describes an indefinitely small space ; the sum of the forces multiplied each [of them by the resolved part, parallel to its direction, of the space described by its point of application, will be always equal to zero ; this resolved part being considered positive when it lies in the dii-ection of its cor- responding force, and negative when in an opposite direction." The resolved parts of the spaces described by the points of appUcation of the forces are called their Virtual Velocities. Let P, Q, H, . . . . denote any system of forces acting on a sys- tem of points consistently with equilibrium ; and let a, j3, y, . . . . denote their virtual velocities ; then, as far as the first poAvers of Oj (i> y> • ■ • • ^^e concerned, Pa+ Q(i + Ry + SS+ =0 (A). The Principle of Virtual Velocities was first detected by Guido Ubaldi' as a property of the equilibriimi of the lever and of moveable pullies. Its existence was afterwards recognized by Galileo" in the inclined plane, and the machines depending upon it. The expression ' moment' of a force or weight acting on any ' Mechanicorain Liber ; De Libra, De Cochlea. ' Del/a Srietizn Meraiiicn, Opera, torn. I. p. 265 j Bologna, 1655. VIRTU AI, VELOCITIE.S. 121 machine, was used by Galileo to denote its energy or effort to set the machine in motion, who accordingly declared that for the equilibrium of a machine acted on by two forces, it is necessary that their moments should be equal, and should take place in opposite directions ; he shewed moreover that the moment of a force is always proportional to the force multiplied by its vir- tual velocity. The word ' moment' was used in the same sense by WaUis\ who adopted Galileo's principle of the equality of mo- ments as the fundamental principle of Statics ; and deduced from it the conditions for the equilibrium of the principal machines. Descartes' has likewise reduced the whole science of Statics to a single principle, which virtually coincides with that of Galileo ; ^t is presented however under a less general aspect. The prin- ciple is, that it requires precisely the same force to raise a weight P tlxrough an altitude a, as a weight Q through an altitude b) provided that P is to Q as J to a. From this it follows, that two weights attached to a machine ^vill be in equilibrium when they are disposed in such a manner that the smaR vertical paths which they can simultaneously describe are reciprocally as the weights. Torricelli^ is the author of another principle which may be immediately deduced from the principle of virtual velocities: the principle is, that when any two weights rigidly connected together are so placed that their centre of gra^dty is in the lowest position which it can assume consistently with the geome- trical conditions to which they are subject, they will be in equi- librium. The principle of Torricelli has given birth to the following more general one, viz. — that any system whatever of heavy bodies will be in equilibrium when their* centre of gravity is in its lowest or highest position. John Bernoulli was the first to announce the principle of virtual velocities under its most general aspect in the form which we have given above, in a letter to Varignon^ dated Bale, Jan. 26, 1717. The striking value of the principle, ' Meehanica, sive de Mofu, Tractatus Geomeiiicus. ' Leltre 73, torn. i. 1657; De Meehanica TractaUis, Opuscu.'a Posthuma. ^ De Hlotu gravium naturaliter descendentium, 1 644. ■* Nouiiclle Mecanique, torn. n. sect 0. 122 VIRTUAL VELOCITIES. as an instrument of analytical generalization, has been splendidly exhibited by Lagrange in his Mkcanique Analytique. From the principle of virtual velocities may be immediately deduced the principle which was proposed by Maupertuis in the Mimoires de V Academie des Sciences de Paris for the year 1740, under the name of the Loi de Hepos ; and which Euler has devel- oped at large in the Memoires de V Academie de Berlin for the year 1751. Suppose that any number of forces P, Q, H, . . . . tending towards fixed centres and functional of their distances p, q, r, . . . . from the centres, to act on a system of points rigidly connected together. Then supposing the system of points to be slightly displaced, so that /;, q, r, .... receive increments dp, dq, dr, .... we shall have, by the principle of "sditual velocities, Pdp + Qdq + Rdr + ....= 0. Let dU denote the left-hand member of this equation ; then (/n = (B). From this it appears that if the system be so placed that 11 may have a maximum or a minimum value, there will be equili- brium : this proposition constitutes Maupertuis' Principle of Kest. It does not however follow conversely that, whenever the sys- tem is at rest, n shall have a maxunum or mininnim value, since by the principles of the differential calculus we know that the equa,tion (B), although a necessary, is not the only condition for the existence of such a value. Lagrange^ has shewn that if U be a minimum the equilibrium will be stable, and if a maximum, unstable. As an example of this theory, it is evident that, if any system be in equilibrium under the action of gravity, there wiU be stable or unstable equilibrium accordingly as the centre of gravity is in the lowest or highest position which is compatible Avith the geometrical relations to which the system is subject. The principle of eqrdlibrium developed by Courtivron^ is like- wise grounded upon the pi-inciple of virtual velocities ; Courtiv- ron's Principle asserts, that if a system of bodies be in motion under the action of any forces varying according to any assigned ' Mecanique Analytique. Premiere Partie, sect. 5. Memoires de V Academie des Sciences de Berlin, 1748, 1749. VIRTUAL VELOCITIES. 123 laws, a position of the system corresponding to a maximum or minimum value of the vis viva will be a position of equilibrium ; a maximum value of the vis viva corresponding to stable, and a minimum to unstable equilibrium. Sect. 1. Equilibrium. (1) A particle P (fig. 76) is attracted towards two centres of force A and B; to find the position of the particle that it may be in equilibrium. Let A, B, denote the two forces ; AP = r, BP = s, AB = a ; draw P3I at right angles to AB, and let A3I = x, PM= y. Then, supposing P to receive some slight arbitrary displacement, the decrements dr, ds, of r, s, will be the virtual velocities of the forces A, B ; hence, by the formula (A), Adr + Bds = (1). But r = (x' + yi, s = {{a - xf + ff, , xdx + ydy - (a - x) dx + ydy ix' + yi ' {{a - xj + fY and therefore by (1), xdx + ydy j^ - (a - x) dx + ydy _ _ (x' + ff {{a-xf-^fY but since dx and dy are independent quantities, whatever be the small variation in the position of P, we have, equating their co- efficients to zero, ^^ _ Bja-x) ^ ^ ^2^^ {x'^yi {(a-xj + y^y ^y + ^ =0 (3). {x^^yi {ia-xJ^fY From (3) we have y = 0, and therefore from (2), we see that A- B; thus it appears that if any particle be acted on by two forces tending towards two fixed centres, the conditions for its 124 VIRTUAL VELOCITIES. equilibrium are, first, that it shall lie in the straight line joining the two centres, and, secondly, that the two forces shall be equal. Euler; Memoir es deVAcacUmie de Berliii, 1751, p. 184. (2) A rigid rod AB (fig. 77) without weight, rests over a peg 0, and against a smooth wall CD, and is acted on by a weight P suspended from the extremity A ; to determine its position of equilibrium and the pressures on the wall and the peg. Draw EOF horizontally ; let AB = a, OB = x, OE = h, AF = y. Let R, S, denote the reactions of the wall and peg against the rod, of which the former will be horizontal, and the latter at right angles to AB. Conceive the rod AB to be slightly displaced from its position of rest by making its end B slide along CD, the peg still touching the rod ; then it is evi- dent that the point B will have no motion parallel to R, and that the motion of the point O of the rod resolved parallel to aS* will be an infinitesimal of the second order. Hence of the three forces P, Q, R, P alone will have a virtual velocity. We have then, by the principle of virtual velocities, Pdtj= 0,ov dij = (1). Now by similar triangles AFO, BEO, there is BE and therefore AF.AO.^, CI U) /I T>\^1 differentiating this equation and performing obvious simplifica- tions, we shall have , a¥ - x^ J dy = ax ; z' {x" - b-j and therefore, by (1), ah'' - .r' = 0, x = {alrf (2), which defines the position of equilibrium. In order to determine S, conceive each point of the rod to receive the same vertical displacement /3, the point B thus sliding along CD and the rod moving parallel to itself. VIRTUAL VELOCITIES. 125 Then, putting ^BOE = .Ff=0 (1); a but since Ap, Aa, are respectively at right angles to OB, OA, it is clear that Z. aAp = Fb or —^ - (p Paa the required condition of equilibrium, from which (p is to be determined. Euler ; Memoires de PAcademie de Berlin, 1751, p. 196. (5) A smooth rod AB (fig. 80) rests against two horizontal bars which pierce the vertical plane through the rod at right angles in the points A' , A" ; the rod passes under the lower and over the higher bar, its lower extremity A being sustained upon VIRTUAL VELOCITIES. 127 a smooth horizontal plane ; to determine the pressures upon the two bars, and upon the horizontal plane. The pressures upon the bars and upon the horizontal plane will be equal to their reactions upon the rod ; the reactions of the bars upon the rod will be two forces R' , K' , at right angles to the rod ', and the reaction of the horizontal plane will be a vertical force R. Let G be the centre of gravity of the rod, at which point we will suppose its whole weight to be collected. Thus we have four forces R, R , R , W, acting respectively at the four points A, A , A% G, rigidly connected together, so as to produce equilibrium. Let AG = a, A' A' = h, and a = the inclination of the rod to the horizon. Conceive the rod to receive a small displacement of such a character that it still remains in contact with the two bars ; then evidently the virtual velocities of W and R will be equal, the one being a positive and the other a negative velocity, and there- fore, a denoting the magnitude of the virtual velocity of each, we have Ea - Wa = 0, and therefore R = TV. (1). Next, conceive the rod to receive a slight displacement, as in (fig. 81), by revolving through a small angle w about the point A' which is supposed to be kept stationary ; the points a, a, g, being the new positions oi A, A' , G ; from a draw mn at right angles to the vertical line through A, and from g draw gn at right angles to the vertical Gji through G. Then, by the prin- ciple of virtual velocities, R . Am - R . A'a' - W. Gn = 0, and therefore, by (1), W{Am - Gn) - R . A'a = (2); but Am = Aa cos a = AA'. w . cos a, and Gn = Gg cos a = AG . w . cos a ; and therefore Am - Gn = AG . o> cos a. - aw cos a, also A'a = A'A". w = bw. Hence from ( 2) we have, substituting for Am - Gn and A'a their values, Wao, cos a - bu) R = 0, 128 VIRTUAL VELOCITIES. and therefore J?' = (3). Again, conceive the rod, as in (fig. 82), to be slightly displaced into the position Aa'ga ; draw gji horizontal and Gn vertical. Then, by the principle of virtual velocities, K. A' a - R. A' a - TV. Gn = ; but Gn = Gg cos a, hence R . A' a - R'. A! a - W. Gg cos a = 0, and therefore, observing i\mtA'a,A'a', Gg, are evidently in the same proportion as AA\ AA', AG, we have R. AA - R. AA' - JV. AG. cos a = 0', and therefore by (3), R. AA = w("^~ ^ AA' + AG cos « = W ^^^ (A A' + b) = W ^-^ . AA; hence R = W ^^^ = R, by (1). (6) A string of given length passes over a given pulley ; it has attached to its two extremities two weights, one of which is capable of sliding freely on a given curve ; to determine the curve on which the other ought to slide in order that in every position of the two weights they may be in equilibrium. Let P, P' , (fig. 83), denote the two weights in any position ; A the pulley ; and let AB be a vertical line through A ; AP = p, AP = p ; m, ni , the masses of the bodies. Draw PM, P'M' , at right angles to AB ; let AM = x, AM' = x' , Z.PAB = ', and we shall get for the equation of the required curve xC/>, 0)=O, X (p, ^) denoting some function of p, (j). John Bernoulli ; Act. Erudit. 1695. Febr. p. 59. Leib- nitz; Ih. April, p. 184. L'Hopital ; Act. Erudit. Suppl. torn. II. sect. 6. p. 289. Fuss ; Nova Acta Acad. Petrop. 1788, p. 197. (7) Four uniform beams AB, BC, CD, DE, (fig. 84), con- nected together by smooth hinges, are placed in a position of equilibrium, the ends A and E being attached to two smooth hinges in the same horizontal line AE; the beam AB is equal to the beam ED, and the beam BC to the beam CD ; to compare the angles BAE and CBD. Let AB = DE=2a, BC=CD=2b, /.BAE=a, ^CBD = (5, AE = c ; h = the height of the centre of gravity of the beams above the line AE; m = the weight of each of the lower and n = that of each of the higher beams. Then (2m + 2}i) li = 2ma sin a + 2n {2a sin a + 6 sin /3) {in + ti) h = {in + 2«) a sin a + nh sin j3 ; but for equilibrium h must be a maximum or a minimum ; hence = (m + 2w) a cos a da^ nh cos j3 f//3 (1). Again, it is evident by the geometry that c = 4a cos a + 45 cos j3, and therefore = a sin a tan a = tan p . n If m = n, we have tan a = 3 tan /3. (8) A beam AB, (fig. 85), rests with one end against a smooth vertical plane OK, and upon a smooth ciu've a/3 ; the plane of the beam and the curve being at right angles to the plane OK; to determine the nature of the curve, that the beam may rest in any position. From any point in the section OK of the vertical plane draw OL horizontal ; from the point of contact P corresponding to any position of the beam draw PM vertical ; let G be the centre of gravity of the beam ; di-aw GH vertical. Let 0M= x, PM- y, AG = a. Then, from the geometry, we see that GH=y-rPG/^ = yJa-.z'^]'If=y-a:f^a'!^. ^ ds ^ \ dxj ds "^ dz ds Xow since for the eqrulibriirm of a material system acted on by gravity, it is necessary that its centi'e of gra%-ity be in the highest or lowest possible position consistent with geometrical relations, it is clear that in the present problem, equilibrium be- ing possible for every position of the beam, GU must be of invariable masjnitude. Hence, if p = — , dx ap C=y-xp + (1 +// difiierentiating with respect to x, and putting (1 -/)' , a hence ;/ = Oj or x = : cJp dx (1 -y/ the former of these solutions gives a straight Hne for the locus of P ; if we integrate the second equation, we shall get, for the equation to the locus of P, VIRTUAL VELOCITIES. 131 Suppose that the origin of co-ordinates be so chosen that when X- a, y = 0, in which case will be the intersection of the beam in its horizontal position with the line OK; then C= 0, and the equation will be a i « T- 3 3 3 X + y = a . (9) A particle is acted upon by three forces A, B, C, passing tlirough thi-ee points A, B, C; to determine the conditions for the equihbrium of the particle by the principle of virtual velocities. The three points A, B, C, must all lie in a single plane con- taining the particle ; also the relative magnitude of the forces A, B, C, are given by any two of the three proportions, A: B:: sin BOC: sm AOC, A: C:: sin BOC: sin AOB, B: C :: sin AOC : sin AOB. Euler; Memoires de VAcademie de Berlin, 1751, p. 185. (10) A particle is acted on by any number of forces; to find the conditions to which their magnitudes and directions must be subject that the particle may be at rest. From the particle draw straight lines representing the forces in magnitude and in direction ; then, that the particle may be in equilibrium, its position must coincide with the centre of gravity of a number of equal particles placed at the extremities of the straight lines. This celebrated theorem for the equilibrium of a particle is due to Leibnitz •} Euler" gave a demonstration by the aid of Maupertuis' Loi de Repos, and Lagrange^ by the Principle of Virtual Velocities. See also Poisson, Traite de Mecanique, tom. I. No. 67. A more general theorem of forces, w^hich com- prehends this of Leibnitz as a particular case, has been given by Chasles:* see BuUetins de VAcademie des Sciefices et Belles- Lettres de Bruxelles, 1840, 2me partie, p. 261. ' Journal des Savans, 1693 ; Opera, torn. in. p. 283. ^ Memoirex de V Academie de Berlin, 1751. ' Mecanique Anahjtique, lorn. I. p. 106. '' Correspondance Mathematique, tom. v. p. 106 — 108; 1829. k2 132 VIRTUAL VELOCITIES. (11) A string of given length passes over a fixed point; it has attached to its two extremities two weights, one of which is capable of sliding freely along an inclined plane passing through the point ; to determine the curve on which the other must be placed that in every position of the two weights they may be in equilibrium. Let the angle which the inclined plane makes with the vertical be a; then, the notation remaining the same as in (6), the equation to the required curve will be (m cos ^ - m! cos a) p = c - nil cos a, which belongs to a conic section. Fuss; Nova Acta Acad. Petrop. 1788. (12) A beam PQ, (fig. 86), rests against a smooth vertical plane AB and a smooth curve AP ; to find the natui'e of the curve that the beam may be at rest in all positions. Let G be the centre of gravity of the beam ; draw P3I hori- zontal ; let PQ = a, GP = c, AM= x, PM= y ; then the equa- tion to the curve will be (c - x f ^i/ _ ^ 2 "*" 2 ~ ' c a which is the equation to an ellipse, the centre of which coincides with Q when PQ is horizontal. (13) A uniform beam AB, (fig. 87), rests upon a smooth hori- zontal plane Ca, and against a smooth vertical plane Ch ; a string ACP is attached to the end A of the beam, and hangs through a small ring at C, with a weight P at its extremity ; to find the position of the beam when at rest. If W denote the weight of the beam, and d the angle BA C, then ^ W (14) A plane figm-e, bounded by a parabola, rests in a vertical plane, on two points in the same horizontal line, the centre of gravity of the figure being in the axis of the parabola at a given distance from the vertex ; to find the position of equilibrium. Let 2a be the distance between the two points, 4m the latus rectum, h the distance of the centre of gravity from the vertex. VIRTUAL VELOCITIES. 133 and 9 the inclination of the axis to the vertical in the position of equilibrium ; then the equation sin 6 {3a" cos* - 4j7i(h - m,) cos^ 9 + 4m'} = will give the positions of equilibrium. (15) A particle is attracted towards each of two fixed centres of force varying inversely as the square of the distance ; to find the equation to the surface on which it may remain at rest in every position. If fx, fjL, be the absolute forces of attraction ; r, r' , simultaneous distances of the particle from the centres ; and a, a , given values of r, r ; then the equation to the surface will be -+-, = - + !-;. r r a a (16) To the extremity ^ of a rod AB, (fig. 88), which is able to revolve freely about A, is attached an indefinitely fine thread BC3I, passing over a point C vertically above A, and sustaining a heavy particle at M on a smooth curve CMN in the vertical plane BAG; to determine the natiu-e of the curve that for all positions of the rod and particle the system may be in equilibrium. Let AB =2a, AC=b, / = the length of the thread BC3I, p = the straight line C3I, 9 = A A CM, m = the mass of the particle, m' = the mass of the rod. Then the equation to the curve will be m'p' + 2 {2?nb cos 9 - ml) p = c, where c is a constant quantity. This problem was proposed by Sauveur to L'Hopital, by whom a solution was published in the Acta Erudttorum, 1695, Febr. p. 56. The curve was shewn by John Bernoulli, lb. p. 59, to be an Epitrochoid. See also James Bernoulli, lb. p. 65. Sect. 2. Stahility and Instability of Equilibrium. (1) AB (fig. 89) is a beam moveable about a hinge ^ ; C is a small pulley in the vertical line thi-ough A, ^C being equal to AG where G is the centre of gravity of AB; a fine string is attached to G which passes over C and has a weight P 134 VIRTVAI. VELOCITIES. suspended by it; to find the stable and unstable positions of equilibrium of the beam. Let GA=CA = a, /=the length of the string GCP, W= the weight of the beam AB, Z. GCA = 9; x = the vertical distance of the centre of gravity of P and the beam below the horizontal Kne thi-ough C. Now, from the geometry, CP = l- 2a cos e ; and the distance of G below the horizontal line through C, is a ^ a cos 2B; hence, by the property of the centre of gravity of bodies, (P + W)x = P(l-2a cos &) + Wa (1 + cos 20). Now for equilibrium x must have a maximum or a minimum value ; hence evidently u = TFcos 20 - 2P cos 6 must have a maximum or minimum value ; therefore ^ = - 2 TFsin 20 + 2P sin = 0, and therefore sin (2 IV cos - P) ^ ; hence for equilibrium it is necessary that sin 6=0, and there- p fore 0=0, or cos = . 2 IF Differentiating u a second time, v,'e get — -, = - 4 ir cos 29 + 2P cos 9 ; ad' if 6 = 0, we have ^ = _ 4 Xr+ 2P; hence -— ^ wiU be positive or negative, and therefore u a mini- mum or maximum according as P is greater or less than 2 TV; hence if P be gi'eater than 2 JV, 6 = gives a position of un- stable equilibrium, and if P be less than 2 JV, one of stability. P Again, if cos 6 = —zrr^, we shall have __ = - 4 Tr(2 cos- 9-\)+2P COS 9= — == — ; da rV VIRTUAL VELOCITIES. 135 if then 2 W be greater than P, -—^ is positive, and therefore the altitude of the centre of gravity of P and the beam is a maxi- mum, and therefore the position will be one of unstable equi- librium ; if 1W ho, less than P, cos will be impossible, or the only position of equihbrium wiU be the unstable one given by 0=0. (2) A uniform beam PQ (fig. 90) is placed upon two smooth incHned planes AB, AC; to find whether its position of equi- librium is one of stability or of instability. Let G be the centre of gravity of the beam ; from P and G draw PMy GH, at right angles to the horizontal plane bAc through A. Let I. BAh = a, L CAc = (5, PG = QG = a, = the angle of incKnation of PQ to the horizon, GH = z. Then, by the geometry, z = a sin + PM= a sin + AP sin a ... ^ sin(i3-0) = a sm u + sm a . 2a -. — 7 rrf sm (a + p) {sin (j3 - a) sin + 2 sin a sin j3 cos 0]; sin (a + /3) then, if z be a maximum or minimum, u = sin (j3 - a) sin + 2 sin j3 sin a cos will be a maximum or minimum ; hence -^ = sin (/3 - a) cos - 2 sin j3 sin a sin = ; do and therefore, for equilibrium, sin (|3 - a) tan (/ = 2 sin j3 sin a a positive quantity, if, as we will suppose, /3 be greater than a. DiiFerentiating u a second time, we have -3^ = - sin (j3 - a) sin - 2 sin (5 sin a cos S ; da from this it appears that, since 6 is clearly less than - , ■— will be negative, or that in the position of equilibrium the centre of gravity is at its maximum altitude; hence the equi- librium will be unstable. 136 VIRTLAL VELOCITIES. (3) A uniform slender rod, acted on by gravity, is placed with its extremities against two planes (one horizontal and the other vertical), having at a point in their intersection an attrac- tive force, varying inversely as the square of the distance, which at the centre of gravity of the rod is equal to half the force of gravity ; to find the position of equiHbrium of the rod, and to ascertain whether it is stable or unstable. Let AB (fig 91) be the rod, G its centre of gravity, P any point in it ; join OP, being the centre of attraction ; draw P3f at right angles to the horizontal line OA. Let AG = a = BG, AP = s, 0P = r, PM= y, L GAB = 9; let the mass of a unit of the rod's length be taken as the unit of mass. Then the attraction on an element ^s of the rod at P will be equal to g^s vertically downwards, and to Ig — Ss towards the centre 0. Hence, adopting the notation which was employed above in the enunciation of jNIaupertuis' Principle, n = 8 W LSs dy + Ig ~ h dA , h and therefore ^11 = g^'^ (8s dy) - \ a'g dt^ — . Now, by the geometry, ^ = 6' sin B, dy = s cos 9 dS ; hence 8'^ (8s dy) = 8"' (sSs cos 9 d9) = S' (sSs) cos 9 d9, and consequently, the limits of the integration being obviously ^> 2a, g-i (gs ^ly^ = 2a- cos 9 dB. Again, by the geometry, we see that r' = s' - 4a cos" 9 s + 4a" cos" 9 ; hence we have 7' (s- - Aas cos" 9 + 4a^ cos^ 0/ = C + log (s - 2« cos" 9 + (s" - 4as cos" 9 + 4a" cos" 9f}, and therefore, between the limits s = 0, s = 2a, ^.1 85 _ , ^ sin 9(1 + sin 9) _ , tan \ (tt + 29) r ~ ^^ ~(^9Tr-cos 9) " ^^ ' tan h 9 " * VIRTUAL VELOCITIES. 137 TT j^i^s \ sec' \ (it + 26/) I sec' i Hence d6 — = " // „/ - " f-rr r tan 1 (tt + 20) tan I 1 1 cos 6 sin Putting for § ' i^sdi/) and c/S ^ — their values in the expression for f/n, we get dll , . f 1 1 — - = i «-(/ + - — - + 4 cos dO \ cos d sni Now there will be equilibrium if n have a maximum or a minimum value, and therefore if sin 6 - cos 0-4 sin 6 cos' 0=0; multiplying this equation by cos 6 + sin 6, a quantity which can- not be equal to zero from = to = ^ tt; we get - cos 20 - (1 + cos 20; (1 + sin 20 - cos 20) = 0, cos 20 + sin 20 + sin 20 (cos 20 + sin 20) = 0, (1 + sin 20) (cos 20 + sin 20) = ; but it is evident that 1 + sin 20 cannot become zero for any value of from to | tt ; hence cos 20 + sin 20 = 0, tan 20 = -1, = i7r, which determines the position of equilibrium. Again, differentiating the expression for — — , we have d-n. , / sin cos f/0- ^ *^ Vcos' snr which is evidently a negative quantity when = g tt ; hence, for this value of 0, IT receives a maximum value, and therefore the equilibrium is one of instability. (4) A square board hangs in a vertical plane by a string, which passing over a smooth nail has its ends fastened to two points symmetrically situated in one edge of the board. To in- vestigate the positions and circumstances of equilibrium. Let G (fig. 92) be the centre of gravity of the board, KCL the string passing over the nail C and attached to the board at 138 VIRTUAL VELOCITIES. the points K, L\ draw GH ?it right angles to KL; let fall KM, HQ, GR, LN, at right angles to the horizontal line passing through C in the plane of the board. Let KL = a, c = the length of a side of the square, / = the length of the string, CK = A, CL = X'; 0, 0', the inclinations of CK, CL, to the vertical, /-HGR = d. Then, from the geometry, putting RG = I u, we have u = c cos + X cos + X' cos ' , , X sm -^ = - cos (9) ; CIA hence, fr-om (6), it may easily be shewn that

= 0', we have 2X sin

; and let m, n, denote the masses of the particles. Then we shall have for the equihbrium, which will be unstable, the equation tani(0-^= tanJ.a. ■ m ~ n (6) A uniform rod passes thi-ough a hole in a spherical shell, and rests with one end against the internal surface, the length of the rod being equal to twice that of the diameter; having given the inclination of the rod to the vertical when it is in a position of stable equilibrium, to determine its inchnations to the vertical when in its positions of unstable equilibrium. If a denote its inclination to the vertical when in its position of stable equilibrium, then its inchnations for its two positions of unstable eqviilibriuni will be 5 (tt + a) and \{jr - a). (7) A pai'ticle is placed in a position of equilibrium between two centres of atti'active force, varpng according to any power of the distance ; to determine for what laws of force the equi- librium is stable and for what unstable. The equilibrium will be stable or unstable according as the forces attract in direct or inverse powers respectively. DYNAMICS. CHAPTER I. IMPACT AND COLLISION. SMOOTH SPHERICAL BODIES. Conceive two spherical bodies, which are composed of the same material, to be moving in the same straight line, namely, in the line joining their centres, and at any time during their motion to impinge against each other. Let tn, m , denote the masses of the two bodies ; and let u, u' , denote their velocities before and v, v' , their velocities after collision ; velocities being reckoned positive in one dhection and negative in the other; then whatever be the absokite or relative magnitudes of the ve- locities u, w', or of the masses m, w', u - u : v - V :: 1 : e, or V - V = e (u - u!) (A), where e is a numerical quantity not greater than unity, which is invariable while the material of the bodies remains the same, but which changes generally with a change in their substance. The bodies are said to be inelastic if e be equal to zero ; im- perfectly elastic if it be equal to any fraction between zero and unity ; and perfectly elastic if it be equal to imity. The theory of collision fiu-nishes us likewise with the follow- ing general relation, m (m - v) = m {p - u') (B). 142 IMPACT AND COLLISION. The signification of the equation (A) is, that the relative velocity of the two bodies after collision bears a constant ratio to the relative velocity before collision, so long as the material of the bodies remains unchanged ; and the equation (B) implies, that the momentum which one body gains by the collision in the positive dii'ection of motion, is equal to that which the other loses. These are the two fundamental principles in the theory of collision. Suppose that u' is equal to zero, and that tn is inconsiderable in comparison with 7n ; then clearly, by (A) and (B), v - V = eu and v = u' = 0, and therefore v = - eu (C), or the small body is reflected backwards with a velocity which is to the velocity of impact as e to 1 ; while the large body experiences no appreciable motion from the collision. This is evidently the case of bodies impinging and rebounding upon the surface of the earth, or upon other bodies firmly attached to it, the earth being regarded as stationary. In the year 1639, J. Marc Marci de Crownland', a Hungarian physician, published at Prague a Avork entitled De Proportione Motus, seu Regula Sphymica, in which he has treated of the collision of perfectly elastic and perfectly inelastic bodies. He occupies himself principally with the consideration of perfectly elastic bodies, and lays down precisely the same rules for their collision which are now commonly adopted. This work, the earliest in which the theory of collision had been correctly pro- pounded, having fallen into general oblivion in the scientific world, the subject was again correctly investigated by the inde- pendent efforts of Wallis, Wren, and Huyghens, Avho apparently had not the slightest knowledge even of the existence of the work by Marci. The laws of the collision of perfectly inelastic bodies were laid down by WalHs, Phil. Trans. 1C68, p. 864, and of perfectly elastic bodies by Wren, Phil. Trans. 1668, p. 867, and Huyghens, Phil. Trans. 1669, p. 925, and Journal cles Sgara?is of March 18, 1669. Wren and Lawrence Rook had, several years earlier than this, exhibited various experi- ' Montucla; Histoire des Mathematiques, torn. ii. p. 406. SMOOTH SPHERICAL BODIES. 143 ments before the Royal Society, in illustration of the principles of collision. The conclusions of Wallis, Wren, and Huyghens, which had been presented to the Royal Society in a very brief shape, were afterwards given more at large by Wallis, Mechanica, Pars Tertia, 1671; Keill, Introductio ad Veram Physicam, Lect. 12, 13, 14; and Mariotte, Traite de Percussion. There are some ingenious experiments by Smeaton on the theory of collision in the Phil. Trans., April 18, 1782. The principles of the collision of imperfectly elastic bodies were first propounded by Newton, Princijna, Lib. i.. Scholium to the Laws of Motion, who inferred experimentally the truth of the equation (A) for any value whatever of e between zero and unity; preceding philosophers having directed their attention to those cases alone in which e is supposed to be either zero or unity. The physical value of Newton's generalization is the more striking when it is considered that natural bodies are never actually endowed with perfect elasticity. For the mathematical formulse in the theory of the collision of imperfectly elastic spheres, the reader is referred to Maclaurin, Clioc des Corps, Prix de V Academie, torn. T. and to Bossut, Cours de Mathematiqice, tom. iii. The results of a series of experiments on the elasticity of bodies, by Mr. Hodgkinson, are to be found in vol. iii. p. 534, of the Reports of the British Association for the Advancement of Science, where he has shewn that the quantity e, in the equa- tion (A), is not, as we stated, and as we shall suppose for the sake of mathematical simplicity, entirely independent of the velocities of the impinging bodies, as Newton had concluded, but that it decreases as the relative velocity increases, assuming however a neaidy constant value when the relative velocity of collision becomes considerable. (1) Two inelastic bodies are moving in opposite directions with given velocities ; to find their velocities after collision. Let m, m , denote the masses of the bodies; a, a, their velocities before collision. Then, putting in the formulae (A) and (B), the direction of the motion of m being taken as the positive one, u = a, ii = - a , e = 0, we have f' - ?? = 0, m (a - ^)) = m (v + a), 144 IMPACT AND COLLTSTOX. and therefore m {a - v) = m' (a! + v), ma - rrid m + m If then ma be greater than ma, the bodies will, after colli- sion, move along in the positive direction "with a common velocity {ina - md) : (tn + m') ; and if ma be less than md, they will move in the negative direction with a common velocity (rdd - ma) : (m! + w?). If ma be equal to m'd, the collision wiU reduce both the bodies to rest. T^^allis ; Mechan. Pars TeHia, de Percussione, Prop. iv. (2) Two perfectly elastic bodies are moving in opposite directions with given velocities ; to find their velocities after collision. The notation being the same as in the preceding example, we have, e being in this case equal to unity, v' - v = a + d, m {a - v) = m! (d + v). Ehminating v from these two equations, ma - ma 2 m'd 7?i + m m + m Eliminating v, we have , md - m'd Ima V m + m m + m WaUis ; Ih. de Elatere et Resilitione, Prop. x. (3) Thi'ee bodies m, m , m' , are placed in a row. The body m receiving a given velocity towards m' , to find the magnitude of ni that the velocity communicated to m by its intervention may be the greatest possible. Let a be the velocity with which m is projected; d the velocity which m' acquires on being struck by m, and d' that which 7n receives on being struck by m'. Then , 277ia ., 2 m' d a = : , a = — z , and therefore 7)1 + 171 m + Amm'a (971 + 77l') {m + 771 ) SMOOTH SPHERICAL BODIES. 145 Since d is to be a maximum, "we must have — , + 1 J w + 7W J a mmimmii ; hence, diiFerentiating with respect to the variable w', — ^, + 1 -jT, (m + m)= m m ' i wi'" - mm = 0, ?n' = (mm")*. Huygheiis; Phil. Trans. 1669, p. 928. Wolff; Elementa Matheseos Unwersce, torn. ii. p. 158. (4) A perfectly elastic sphere impinges with a given velocity and in a given direction against a smooth plane ; to determine the velocity and direction of reflection. Let u, V, denote the velocities of incidence and of reflection, and a, j3, the angles which the directions of the motion before and after impact make with a normal to the plane. The resolved parts of the velocities parallel to the plane are u sin a and v sin j3, and, at right angles to it, u cos a and v cos j3. But the plane being perfectly smooth will not affect the resolved part of the velocity parallel to itself, and therefore V sin j3 = M sin a ; while the other resolved part of the incident velocity will be affected as if the impact had been direct, and therefore, by ( C), V cos /3 = w cos a. From these two equations it is evident that tan j3 = tan o, j3 = a, and v = u, or the angle of reflection is equal to that of incidence, and the velocity of reflection to the velocity of incidence. "Wallis; Mechan. Pars Tertia, De Elatere, &c. Prop. ii. (5) Two smooth spheres moving with given velocities and" in any given directions whatever, impinge against each other ; the spheres being supposed perfectly elastic, to determine their velo- cities and the directions of their motions after collision. L,etAB, A'B', (fig. 94), be the directions of the motion of the two bodies before collision ; and O, O , the positions of their L 146 IMPACT AND COLLISION. centres at the instant of contact ; produce 0' O indefinitely to a point C. Let a, a', denote their velocities before collision, and a, a, the angles AOC, A'O'C. Then the resolved parts of the velocities of the spheres 0, 0', in the direction COO' will be a cos a, a cos a, and at right angles to 00' in the planes AOC, A'O'C, respectively, a sin a, a sin a. These latter resolved velocities will not be affected by the collision. The former will be affected exactly as if the sphere moving along COO' with a velocity a cos a were to impinge directly upon the sphere 0' moving with a smaller velocity a cos a estimated in the same direction. Hence if v, v', denote the resolved parts of the velocities after collision parallel to the line COO', we have, as may be readily ascertained by the principles of this chapter, ma cos a - m'a cos a 1m a cos d V = + _ , m + m 7)1 + m , m'a cos a - ma cos a 2ma cos a V = ; +. J-. m + m 7n + m Let V, V , denote the velocity of the spheres 0, 0', after colli- sion, and (p, (j>', the angles which the directions of their motions make with 00' ; then V^ = v'' + a^ sin' a, V' = iP + a'^ sin" a , a sin a , a sin d tan (h = , tan = ; , V V their motions still taking place in the planes BOO, BOO. Keill; Introductio ad Veram Physicam, Lect. 14. (6) Two imperfectly elastic bodies are moving in the same direction along the same straight line Avith given velocities ; the one overtakes the other and collision ensues ; to find the veloci- ties of the two bodies after collision. If 171, 711 , be the masses of the two bodies, e their common elasticity ; a, a! , their velocities before, and v, v , their velocities after collision ; 7na + 7n'a' em' (a -a) v = ~- , r^, 7n + 771 7n + 7n , ma + rna! em (a - a') V = — + ^^ ^ . 7n + 7n m + 7n Maclaurin; CJioc des Co7~ps, p. 30, Piix de VAcademie, tom. i. SMOOTH SPHERICAL BODIES. 147 (7) To find the sum of the vires vivce of two perfectly clastic bodies after direct collision. If a, a, be the velocities before, and v, v , after collision, mv' + m'v' = ma' + tn'a'. Huyghens ; De Motu Corportnn ex Percuss. Prop. xi. •John Bernoulli; Discours sur le Moucement, chap. x. (8) To find the sum of the vires vivce of two imperfectly elastic bodies after direct collision. The notation remaining the same as in the preceding example, and e denoting the elasticity, 2 , „ , , ,2 (1 -e')mm{a - a J mrr + m v = mcr + ma - — ; , m + m which shews that vis viva is lost by the collision. (9) To find with what velocity a ball must impinge upon another equal ball moving with a given velocity, that the imping- ing ball may be reduced to rest by the collision, the common elasticity of the balls being known. If e be the common elasticity, and a the velocity of the ball which is struck, the impinging ball must impinge with an op- posite velocity ecjual to 1 + e a. 1 - e (10) To determine the velocities of two bodies A and B of given elasticity and given masses moving in the same direction, that after collision A may remain at rest and B may move along with an assigned velocity. If m, ni! , be the masses of A, B, e their elasticity, /3 the velo- city which B is to have after collision, and a, h, the required velocities of A, B, before impact, 1 + e m'fi J (em' - m) B a = ; , b = — ;- . e m + ni e {m + m ) Maclaurin; Choc des Corps, p. 52, Prix de VAcad. torn. i. l2 148 IMPACT AXD COLLISION. (11) A spherical body A impinges directly with a certain velocity upon a spherical body B at rest, the common elasticity of the two bodies being given ; to find the mass of a third body which, moving with the velocity which A has before the collision, shall have the same momentum which B has after the collision. If tn, 771 , denote the masses oi A, B ; e the common elasticity of A, B ; and m the mass of the required body, ., , mm' m = ( 1 + e) — J . m + 771 (12) To find the elasticity of two spheres A and B, and the ratio between theh masses, that when A impinges upon B at rest, A may be reduced to rest, and B move on with the w*^ part of ^'s velocity. If 771, 171 , denote the masses of ^, B, and e their common elasticity, then 1 17l' e = - , — = 71. n 7)1 (13) Two perfectly elastic spheres meet directly with equal velocities ; to find the relation betv>^een their magnitudes, that after collision one of them may remain at rest. If 171, m, denote theh masses, m corresponding to the one which remains at rest, m' : m :: 3 : 1. (14) Any number of spheres of given elasticity being ar- ranged in a straight line, and one of the extreme ones having a given velocity communicated to it so as to bring it into direct collision with the adjacent sphere of the series ; to determine the velocity ultimately acquired by the last sphere. If r be the number of the spheres, e their common elasticity, m,, m,, m^, m^, their masses, and a the velocity WiXh. which the first is projected ; then v being the velocity acquired at last by m^, V = (\ + ej ^ — . — . ?— . a. m^ + m, m^ + 771^ m^ + m^ nir + m^^^ Maclaurin ; Choc des Corps, p. 54, Prix de V Acadhnie, torn. i. SMOOTH SPHERICAL BODIES. 149 (15) A number of equal spheres are placed on a smooth table in a straight line and close together ; they are connected together by equal inelastic threads ; a motion is given to the first in the direction of the line which they form so as to separate it from the second ; to find the time which elapses before the last sphere is put in motion. If n be the number of spheres, a the length of each of the connecting threads, and /3 the velocity with which the first sphere is projected, then . , w ()i - 1) a time requu'ed = 7=; . ^ 1 . 2 j3 (16) Between two spheres of given masses is placed a row of spheres ; a velocity is communicated to one of the original spheres so as to bring it into direct collision with the nearest of the intermediate ones ; to find the requisite magnitudes of the intermediate spheres in order that the velocity acquired by the last sphere may be the greatest possible, and to determine this velocity when their number becomes indefinitely great. The intermediate spheres must be geometrical means between the two original ones ; and if ?n, tn, denote the masses of the original spheres, a the velocity communicated to m, and a that acquired by m when the number of the intermediate spheres becomes infinite. (17) An imperfectly elastic sphere impinges upon a plane; to find the angles of incidence and of reflection, that the velocity before may be to the velocity after impact as 2^ : 1, the elas- ticity being equal to — . Angle of incidence = g tt, angle of reflection = 5 tt. (18) An inelastic sphere A, moving with a given velocity, impinges upon an inelastic sphere B at rest, the line joining the centres of the two spheres at the instant of collision making a given angle with the direction of ^'s motion ; to determine the velocity of A after collision. 150 . IMPACT AND COLLISION. If a be the given angle ; m, m , the masses of the spheres A, B; and a, v, the velocities of A before and after collision, r = a < sm" a + j —, cos o \ . y [m^my J (19) A sphere A in motion is struck by an equal one B mov- ing with the same velocity, and in a direction making an angle a ■svith that in which A is mo-^-ing, in such a manner that the line joining their centres at the time of impact is in the direction of ^'s motion ; to find the velocities of the spheres after impact, and to determine for what value of a that of A will be a maxi- mimi, the common elasticity of the spheres being supposed to be known. If a denote the velocity of each of the spheres A, B, before, and u, V, their respective velocities after impact, then, e being their common elasticity, u' =\ (f {l ^ e + (I - e) cos a}' -^ a" sin'^ a, v' = la' {1 ^ e-(\ -e) cos a}". 1 - e A\'Tien w is a maximum, cos a = . 3 - e (20) Thi'ee perfectly elastic spheres A, B, C, are placed at the three angles of a plane triangle of which the angles are known ; to compare the magnitiides of the spheres, when A impinging obliquely upon B is reflected so as to strike C, and thence reflected to its first position; the lines joining the centres of the spheres A, B, and A, C, during collision, being respec- tively perpendicvJar to the opposite sides of the triangle, and their diameters being inconsiderable in comparison with the sides of the triangle. If m, m , m , denote the masses of the spheres A, B, C; and a, /3, y, the angles of the triangle at which A, B, C, are placed, ni _ sin /3 m sin y m sin (a - y) ^ m sin (/3 - a) ' (21) A sphere A (fig. 95) mo\-ing in the direction ^^i^^^ith an assigned velocity impinges upon a sphere B at rest, the two spheres ha^dng the same elasticity; supposing AF' to be the SMOOTH SPHERICAL BODIES. 151 direction of ^'s motion after impact, and KABL to be a straight line passing through the centres of the two spheres at the time of collision, to find the value of the angles EAK and FAF' when the latter angle has its greatest value. If 71 be the ratio of the mass of A to that of B, e be the com- mon elasticity of the spheres, L EAK = 9, I. FAF' = ^ ; then 1(1 +e) {(n + l)(n- e)Y ^^^ ^ ^ ( ;rT"l I ' ^'''' "^ ^ ( 152 ) CHAPTER 11. RECTILINEAK MOTION OF A PARTICLE. The determination of the circumstances of the motion of a material particle, which moves in a straight line under the action of a finite accelerating or retarding force, depends upon the two following differential equations, called the equations of motion of the particle dx civ J, where t denotes the time of the motion reckoned from an as- signed epoch, X the distance of the particle at the end of this time from an assigned point in the line of its motioil, v the velo- city, and y the accelerating or retarding force. From these two equations we readily deduce the two following, cPx J, dv ^ ~df^^' ""Tx^^- These equations, which constitute the complete expression of the circumstances of rectilinear motion in the language of the dififerential calculus for every condition of acceleration or retar- dation, are due to Varignon, and were published in the Mem. de VAcad. des Sciences de Paris, 1700, p. 22. It maybe observed however that, long before this, geometrical investigations of rec- tilinear motion for variable forces had been given by Newton.' From the formula vdv = fdx we see that dv' varies as fdx : an opinion however was expressed by Daniel Bernoulli,^ that there is no reason to consider this the only possible law of variation ; for instance, that we might as well have dv" oc fdx, n being any quantity whatever. In opposition to Bernoulli's suggestion, Euler^ endeavoured to prove that the law of the sqviare of the ' Principia, lib. i. sect. 7; lib. ii. sect. 1. ' Comment. Petrop. 1727, p. 136. •' Mechanica, tom. i. p. fi2 et se(]. RECTILINEAR MOTION OF A PARTICLE. 153 velocity is necessarily true ; and D'Alembert^^sliewed the truth of this law to depend simply upon the definition of the meaning of the symbol y. The complete solution of a problem in rectilinear motion con- sists in the determination of relations between every two of the quantities x, v, f, t: now the general equations of rectilinear motion furnish us with only two independent relations between these four quantities ; it is evident then that the data in every problem must consist in the expression of some particular equa- tion, ^ {x, V, f,f)=0 between x, v, f, t, so that we may have, in all, tlu-ee equations connecting the four variables. The function {x, v,f, t) may involve two, three, or all of the quantities x, v, f, t ; and by the theory of combinations it is evident that there will be six varieties of the first, and four of the second class ; hence the general problem of rectilinear motion resolves itself into eleven distinct classes of problems. We shall however confine ourselves to the consideration of those two classes in which the given function involves either x, f, alone ; or X, f, V, alone : under the former head we shall exempUfy the motion of a particle in vacuo ; under the latter, in a resisting medium. The other classes are devoid of any physical interest. Sect. 1. Motion in Vacuo. (1) A particle is placed at a centre of repulsive force which varies as any power of the distance ; to determine its velocity after receding to any distance from the centre, and the time of the motion. Let fx represent the absolute force, x the distance of the par- ticle from the centre of force after a time t, and v the velocity. Then, for the motion, we have dv V -^ = ux". ax Integrating with respect to x, and bearing in mind that v = Q when X - 0, we have i d'^pl] X" J dx - w + 1 ' Traite dc Dynamiqne. 154 RECTILINEAR MOTION OF A PARTICLE. and therefore v"^ = — — x"*\ n+ I which gives the velocity for any value of x. Again, ^=. = f-^>>-): hence, t being equal to zero when x =^ 0, there is 2 fl^^^\l^^,y X'^ 1 - w \ 2^ Euler; 3Iechanica, torn. i. p. 123. (2) A particle being attracted by a force varying inversely as the ri}^ power of the distance, to find the value of 7i when the velocity acquired from an infinite distance to a distance a from the centre is equal to the velocity which would be acquired from a to \ a. Let ^ denote the absolute force, x the distance of the particle from the centre of force after a time t, and v^, v^, the two veloci- ties. Then, for the motion of the particle, dv _ fi clx x"- ' Hence, for the former motion, v being equal to zero when X = (X>, 25,^ = - 2u I — dx= -. ^ — -^ , in-X) and, for the latter motion, since « = when x = a, But by hypothesis v^^ is equal to v^ ; hence 2/x 1 2^ ir(4"-'-a w - 1 a""^ w - 1 a and therefore 1 = 4""' - 1, 4""' = 2 ; hence n ~ 3. RECTILINEAR MOTION OF A PARTICLE. 155 (3) Four equal attractive forces are placed in the corners of a square, their intensity varying as any function of the distance ; a particle is placed in one of the diagonals of the square very near to its centre ; to find the time of an oscillation. Let O be the centre of the square (fig. 96), E the position of the particle after any time t from the commencement of the mo- tion ; let OD = a, OE = x, AE = CE = r. Then, for the motion of the particle, taking the sum of the forces acting upon it in the line OD, we have ~ = ~ 2(j> (r) ^ + (a- X)- (t>(a^ x), and therefore, neglecting powers of the small quantity x higher than the first, we get, by Taylor's theorem, or, putting the coefficient of x equal to - 2k, cVx „, df . . die Multiply both sides of this equation by 2 ~ , and we get by integration df where Cis an arbitrary constant. dx . . . . Let j3 be the initial value of x ; then, -j- being initially equal to zero, we have = C - k^\ dx: If dof and therefore -— - = h (j3' - oi?') ^ I dx dt = J^' (ii'-xj' the negative sign being taken because x decreases as t increases. Integrating, we get no constant being added because x = (3 when ^ = ; hence ^ = /3 cos (k'' t). 156 RECTILINEAR MOTION' OF A PARTICLE. _i NovT as soon as ^" t becomes equal to tt, x becomes equal to - /3, its greatest negative value. Hence the time of a complete oscillation being T, we have li' T = TT, and therefore, substituting for k its value, (4) A particle A attracts a particle B with a force always to that with which B attracts A in the ratio of fi to fi ; the parti- cles bemg originallv at rest, to find their position as well as that of their centre of gravity after any time ; the intensity of each force being directly as the distance between the particles. Let be a fi:s;ed point in the line of the motion of the particles, (fig. 97), and let OA = x, OB = x , at any time t. Then, for the motion, we have d'x / , . / X 7^=^^-"-^^ ^'^' d'^x -^ = -fj!{x'-x) (2). Multiplying (1) and (2) by fx and by fx respectively, and add- ing the results, we get , d'x d'x ^ df ^ df dx dx Integrating, and bearing in mind that -r^ , -y- , are both equal to zero initially, ^^ ^/^' integrating again, fxx ^ fix = fi'a -^ fia (3), a, a', being the initial values of x, x. Again, subtracting (1) from (2), d^ -, (x -x)^(ji^ fi') (x - x)=0. df and putting x - x = z, RECTILINEAR MOTION OF A PARTICLE. 157 multiplying by 2 -r^ , and integrating, dz~ -^(^^^)z^=C, '\iexe Cis an arbitrary constant. But, initially, dz , , — = 0, and z = a -a, (tt and therefore {fi + f^) {a - aj = C; aj- -'}. dt 1 1 dz~ , ,i ■ r/ _ , -A2 1 ' ..2T5 {fx + fjif [{a'-af-z^Y the negative sign being taken because ^ or x - x decreases as t increases. Integrating, we have 1 , z 1 ,x'-x t = — — — cos -; = cos — , {jx + n) (/* + /^ ) no constant being added because x - x= a - a when ^ = ; hence x - x = (a - a) cos {(p. + ^u')* t] (4). From (3) and (4) we readily obtain ua + ua u(a' - a) , , , A . X = '— , ~ '-^-; ^ cos {(jU + ^' t\, fi + fj n + fi , fi'a + j.ia' fi (a! - a) (m + m') X = mx + mx cos K/x' 4 /// t\ ; (fx'a + ua') + ^ , ^ (a - a) cos {{u + fxj t] , h+H- h + H^ where m, m', denote the masses of A, B, and x the distance of their centre of gravity from at any time t. If /i', /i, be proportional to m, 771 , respectively, then clearly from our general result , ,, _ m + m , I ,. [in + m)x = — , {fxa + /Jia) fi + H _ u'a + ua' or X = , - , fi + fX 158 RECTILINEAR MOTION OF A PARTICLE. which shews that the centre of gravity remains stationary during the whole motion. (5) A body not affected by gravity falls down the axis of a thin cylindiical tube infinite in length, the particles of which at- tract with a force which varies inversely as the square of the distance ; to find the velocity acquired in falling through a given space. Let k be the thickness of the tube, r the radius of its interior surface, x the distance of the particle P from the extremity of the tube after a time t ; then the volume of a portion of the tube con- tained between slices at distances s and s -^ ds from P will be 2Trrk(Is, and therefore the attraction of this elemental portion on the particle along the axis of the tube will be, the unit of attrac- tion being chosen to be the attraction of a unit of mass at a unit of distance, \ 5 2wprkds . , — ^. , ^'^'' (s' + ri where p denotes the density; and therefore for the motion of the particle we have d^x , f" sds = 2;r^rA-. J = ^: -^ (sUrj' (x' + rj multiplying by 2 — and integrating, dx'^ - ^^= -j^ = ^irprk log [x + (x' + rf] + C. But t' = when x= Q ; hence = 'iTTprk log r + C, and therefore t-' = Airprk log ^ + (^' + ^^ , r (6) A particle is projected downwards with a certain volocity on a horizontal plane thi'ough a height of twenty feet ; it re- bounds ten feet and then falls again and rebounds four feet : to find the elasticity of the particle and the velocity of projection. RECTILINEAR MOTION OF A PARTICLE. 159 Let a denote the velocity of projection, and e the elasticity ; then if t? denote the velocity at the end of the first fall just before impact, we shall have V- = a + 40g (1). The velocity at the beginning of the first rebound will be ev, and since this velocity is lost by an ascent through ten feet, there is e"r= 20<7 (2). Again, since e" v, the velocity at the beginning of the second rebound, is lost by an ascent through foiu* feet, we get ^^=8g (3). From the equations (l), (2), (3), we readily see that e = 0J, a = (lOc/)K Taking for g its approximate value 32i feet, we obtain 18 feet nearly for the value of a. (7) A particle falling in a straight line towards a centre of force, the intensity of which varies as the w'^ power of the dis- tance, acquires a velocity /3 on arriving at a distance a from the centre ; to find at what distance z from the centre of force it must have commenced its motion. Let ju denote the absolute force ; then z will be given by the equation Euler; Mechan. tom. i, p. 109. (8) A particle falls towards a centre of force, of which the intensity varies inversely as the cube of the distance ; to find the whole time of descent. Let fx denote the absolute force and a the initial distance ; then time of descent = — . I 4 (9) A particle descends from an infinite distance towards a centre of force which varies inversely as the square of the dis- tance ; to find the velocity at a given distance from the centre of force. 160 KECTILINEAK MOTION OF A PARTICLE. Let fi be the absolute force and a the given distance ; then velocity = | — J. (10) A particle is placed at an assigned point between two centi-es of force of equal intensity attracting directly as the dis- tance ; to determine the position of the particle at any time, and the period of its oscillations. Let a denote the initial distance of the particle from the middle point of the line joining the two centres of force, x the distance after the expii-ation of a time t, and /j. the absolute force of each centre. Then 1 _ . . TT X = a cos {(2fxft}, and period of an oscillation = . (11) A particle acted upon by two central forces, each attracting with an intensity varying inversely as the square of the distance, is projected from an assigned point between them towards one of the centres; to find the velocity of projection that the parti- cle may just arrive at the neutral point of attraction and remain at rest there. Let ju, fx, denote the absolute forces of the two centi-es ; 2^/, 2a', the initial distances of the particle from the two centres ; and F'the velocity of projection. Then i \ F- = ^ + ^ - {fi_tJL^' a a a + a (12) A centre of force C (fig. 98) moves along the straight line OA Avith a imiform velocity, attracting, with a force varying directly as the first power of the distance, a particle P which is moving in the same straight line ; having given the initial posi- tion of C, and both the initial position and the initial velocity of P, to find the position of P at any time. Let a, a, be the initial distances of C, P, from O ; /3 the uni- form velocity of C, and /3' the initial velocity of P ; x the distance of P from O after a time t ; fi the absolute force of attraction. Then x = a + (5f + ^-~ sin {^H) + {a - a) cos (jit). i Eiccati ; Bonon. Institut. tom. vi. p. 138; 1783. RECTILINEAR MOTION OF A PARTICLE. 161 (13) The circumstances remaining the same as in the pre- ceding problem, except that the force is repulsive ; to find the position of P at any time. 2fi^ 2fx' Riccati; /Z». p. 151. (14) Supposing the centre C to move along OA with a uniform acceleration, attracting directly as the distance ; to determine the place of P at any time. Let f represent the increment of C"s velocity in each unit of time, and j3 its velocity at the commencement of the motion ; then, the notation remaining the same as in the two preceding problems, x=.a-i-+Qt + Iff + ^-^ sin {ix't) + ( a' - a + "^ ) cos (j^H). A* Riccati; Ih. p. 168. (15) The circumstances and notation remaining the same as in the preceding example, except that the force is repulsive ; to find the place of P at any time. 2^^ «-«■+•'- l+(/3-i3')r r J /«_«'+/ )_(^3_^') -M ^ ^ -^ ^ 2,x; Riccati; Ih. p. 182. (16) A particle is placed at a given distance from a uniform thin plate of indefinite extent, every particle of which attracts MT-th a force varying inversely as the square of the distance ; to find the time in which the particle will arrive at the surface of the plate. Let k denote the thickness of the plate, p its density, and a the initial distance of the particle from it. Then ( « V time = — ■:. I . \TrpkJ M 162 RECTILINEAR MOTION OF A PARTICLE. (17) A particle is placed at a small distance from the centre of a thin ring of uniform density and thickness, every molecule of which repels with a force varying inversely as the square of the distance ; to determine the position of the particle at any time, and the period of its oscillations. Let / be the initial distance of the particle from the centre of the ring, a the radius of the ring, k the area of a section, p the density, and x the distance of the particle from the centre at the end of a time t. Then, the repulsion of a unit of the ring's mass at a unit of distance being taken as the unit of repulsion, X -I cos \ — ^-— ^ t and period of an oscillation = ( -7 | a. I a J \ph) (18) A particle being placed at a given distance from a thin circular lamina of uniform density, in a line passing through its centre and perpendicidar to its plane, to find the velocity which it will acquire by moving to the circle, the attractive force of each molecule of the circle varying inversely as the square of the distance. Let a be the radius of the circle, h its thickness, p its density, h the given distance ; then, the unit of attraction being the at- traction of a unit of mass at a unit of distance, and V being the velocity required, V = ^irph {« + &-(«'' + hj}. (19) A body of known elasticity falls from a given altitude above a hard horizontal plane, and rebounds continually till its whole velocity is destroyed ; to find the whole space described. If a denote the first altitude, e the elasticity, and s the re- quired space, 2 (20) Two perfectly elastic balls beginning to descend from different points in the same vertical line impinge upon a per- fectly hard plane inclined at an angle of 45'', and move along a horizontal plane with the velocities acquired ; to find what dis- tance they will move along the horizontal plane before collision takes place. RECTILINEAK MOTION OF A PARTICLE. 163 If a, a , denote the altitudes through which they fall, and s the distance required, , ,,^ ^ s = {adj. (21) From a point ^ in a vertical line AB falls a particle from rest; at the same instant another particle is projected upwards from B with a given velocity : to find when and where the two particles wUl meet ; the motion being supposed to take place in a vacuum, and gravity being the only force to which the particles are subject. Let a be the length of the line AB, j3 the velocity of projec- tion of the ascending particle, x the distance from A at which collision takes place, and t the time of this event from the com- mencement of the motion. Then qci' , a T — K- t — — 2/3'-' ^~^- Kurdwanowski ; Mem. de VAcad. des Sciences de Berlin, 1755, p. 394. Sect. 2. Motion in Resisting Media. The retardation experienced by a material particle in travers- ing a resisting medium of variable density, depends at any point of its path upon the density of the medium and the velocity of the particle, and will therefore be some function of these quantities. The nature of this function can be ascer- tained only by experiment. In mathematical investigations, for the sake of simplicity and as a probable approximation to the truth, the function is assumed to be of the form kpQ, where p denotes the density of the medium and ^ some function of the velocity of the particle ; and where h is an invariable coefficient depending upon the nature of the particular medium in respect to the tenacity and the friction of its constituent molecules. For the earliest mathematical development of the theory of the resistance of media to the motion of bodies, we are indebted to the labours of Newton and Wallis. The profound researches of Newton on this theory were pubHshed in the year 1687 in the second book of the Principia. In the same year, after the M 2 164 RECTILINEAR MOTION OF A PARTICLE. publication of Newton's investigations, Wallis, who had indepen- dently arrived at valuable conclusions on the subject, communi- cated his reflections to the Royal Society, which were inserted in the Philosophical Transactions for the year 1687. There is a paper by Leibnitz on the question of resisting media in the Acta Erudit.Ijips. ann. 1689, in which he developes opinions which he declares to have been communicated by him twelve years before to the Royal Academy of Sciences. Huyghens also has dis- cussed certain points of the theory at the end of his Discours de la Cause de la Pesanteur, published in the year 1690. Finally, all which these philosophers had communicated to the scientific world either with or without demonstration, was investigated analytically by Varignon in a series of papers in the Memoires de VAcad. des Sciences de Paris, for the years 1707, 1708, 1709, and 1710. There is an elaborate paper by Bouguer in the Mem. de VAcad. des Sciences de Paris, 1731, p. 390, in which he investigates the motion of a particle in resisting media which are themselves in motion. (1) A particle acted upon by no forces is projected with a given velocity in a resisting medium of uniform density, where the resistance varies directly as the velocity; to determine the velocity and the space described at the end of any time. For the motion of the particle we have dv "S-""' where ju is some constant quantity ; hence dv dx = -^' V = C - fix. LetjS then denote the initial velocity when (5 = C, and therefore V = (5 - fix; X is supposed to be zero; whence, v being equal to dx 'di' we have (5- fiX RECTILINEAR MOTION OF A PARTICLE. 165 ^ = C - - log ()3 - fiX). But ^ = when x = ; aiid therefore 0=C-ilogi3, hence t = - log ""^ fi (3 - fXX ,u. i3 and therefore v = jSe"/^'. Newton ; Principia, lib. ii. Prop. 1 and 2. Leibnitz ; ^cto Erudit. Lips. ann. 1689. Varignon; Mem. de r Acad, des Sciences de Paris, ann. 1707, p. 391. (2) A body faUs towards a centre of force which varies as the inverse cube of the distance, in a medium of which the density varies also as the inverse cube, and of which the resistance varies as the square of the velocity; to find the velocity at any dis- tance from the centre. Let X represent the distance of the particle from the centre after a time t, and let a be the initial distance. Let k denote the force of resistance at a unit of distance for a unit of velocity, and n the absolute force of attraction. Then, for the motion of the particle, d^x fjt k dy? 'de^~J^^x^de' dx d'^x 2/u dx Ik da? Ht lie ^ ~ ~x^ It ^ ^' de ' d dx^ d 1 , dx' d 1 dtde^^Jt?~ dedtx^' Assume ^-^ = w and — = z; then de 7? die dz , dz — - = ^ — — kiv -^ , dt dt dt 166 RECTILINEAR MOTION OF A PARTICLE. dw + kiodz = fidz, d {t'^tv) = 11^' dz, k Hence, putting for xo and z their values, - 1 f'^ tj' = c + 1 f\ But X = a when v = ; hence k 0= c+^?, k k^ k k When X becomes equal to infinity, suppose V to be the value of ti : then „ * r^ = ^ (1 - ,'?\ V is called the terminal velocity of the particle, a technicality invented by Huyghens\ to signify the ultimate velocity of a par- ticle descending in a resisting medium to an indefinitely great depth. (3) To determine the motion of a particle, not acted upon by any force, when the resistance varies as any power of the velocity. For the determination of the relation between the velocity and the space, V ~- = - kv", dx J ^ dv kdx = , kx = C + ^ {n - 2) t)"-' Let a; = and « = j3 initially ; then ' Discours sur la Cause dc la Pesanteur, p. 170. RECTILINEAR MOTION OF A PARTICLE. 167 (n-2)kx=±,--^, (1). Again, for the relation between the velocity and the time, dv , , , dv dt=-^''' kt= C + {n But tJ = j3 when ^ = ; hence (»-l)^-« = ^,-p (2)- If between (1) and (2) we eliminate v, we shall obtain a relation between s and t. Varignon; Mem. de fAcad. des Sciences de Paris, 1707, p. 404. (4) A particle acted on by gravity falls from a given altitude in a medium of uniform density, where the resistance varies as the square of the velocity ; on arriving at the lowest point of its descent it is reflected upwards with the velocity which it has acquired in its fall; after reaching its greatest altitude it again descends and is again reflected ; and so on perpetually : to de- termine the altitude of ascent after any number of reflections. Let the maximum altitudes of the particle be represented by a^, a^, a^,....a^ being the altitude from which it originally falls. Let c denote the volume of the particle, and p, p, the density of the particle and of the fluid. For the descent down any of the altitudes there is dx cp V — =q' - kv^, where o^' = ( 1 - - dx " \ P. vdv J 7-2 = ^^' g - Kv -^log(^' -M) = X+C; but, the origin of x being the highest point. or 168 RECTILINEAR MOTION OF A PARTICLE. hence — loa: — = z, and therefore, if v^ denote the velocity acquired down the w'^' altitude, log = 2ka^, 1 ,v 9 " -,v'=\-t'""'- (1). 9 " For the ascent up the {n + 1)'** altitude, the origin of x being the lowest point, l\o^{g'^-kv:)=C, — log^' = C-a„,,, llog(l+io = «„., %v' = 6'*"-' - 1 (2). 9 " Hence, from (1) and (2), j^Kifi + £-'K = 2: assume 6^*"" = u„, and we have w ^, + — = 2, Putting w^ = t)^ + 1 , we get K + l)(^Vi+l)+ 1 = 2(«„+ 1), V V n n 1 ^ , 1 w+l+C = n + 6, ?/„ - 1 = — — Ty* K = ?<-l " n + C " n + C RECTILINEAR MOTION OF A PARTICLE. 169 But r = 1 -f C; hence Wj - 1 1 n + M, - 1 nu.-n^r\ U = ' = ;; \ 1 m - 1) w, - w + 2 e^, - 1 Or, putting for u^^, u^, their values, 2ka., m'^''"' - n + 1 a (n - 1) £'*"' - n + 2' 1 , WE^*"' - /^ + 1 « = 777 log 2A ^ (w - 1) 6'*"' - w + 2 ' If fltj be equal to infinity, 1 , 71 a = -^loi 2k '^ n - 1 Euler; Mechan. torn. i. p. 192. (5) To determine the centripetal force that a particle may always descend to a given centre in the same time from whatever distance it commences its motion ; the density of the medium in which the particle moves being known at every point in its path, and the resistance varying as the square of the velocity. The equation of motion is dv , 2 V -^ = - p + kv , ax where p denotes the centripetal force, and k the density at any point. Multiplying by 2£-'/*''' dx, the equation becomes die-'^"'' v') = - 2 e'^^'"''pdx. Integrating, e''-''*''' v^ = C - 2f£---^'"'''pdx. Let a denote the initial distance of the particle from the centre of force ; then, the velocity being initially equal to zero, where A ^ 2 I ^•■^'""'pdx and X = 2 I i'^-^ '"'' pdx. Jo Jo 170 RECTILINEAR MOTION OF A PARTICLE. Therefore v = (A- xf a^'"', t'f'"' dx at = . {A-Xf ^ow since X, k, are both functions of x, it is clear that we may assume ^X t^'^'' dx = — where P is a function of X alone ; hence Again, eliminating /^between the equations (1), we have d^y d'z de ^ dt' FREE CURVILINEAR MOTION OF A PARTICLE. 1T7 integrating, and adding a constant c which will represent twice the area described in a unit of time about the centre of force, we ^ dt " dt and therefore, by (3), hence, from (4), \1mx - y')-r- = 2mc ^ ^ dt „ 4m"c^r cV , .^, JP = = bvr2i (2}7ix - yy 2m (2m - xf ' ^ ^ ^' (3) A particle P, moving with a uniform angular velocity round a fixed point 0, is acted on by a force always perpendi- cular to the straight line OP ; to find the polar equation to the orbit, being the pole, and the particle having initially no velocity in the direction of the radius vector. Let the axis of x coincide with the prime radius vector ; L POx = B, OP = r, F ^ the force, w = the invariable angular velocity. Then for the motion of the particle there is d'^x _ Fy d^'y _ Fx 'df^'"V' 'df ~ V' hence we have d'x d'lj ... :r— - + y^ = (1). df -^ df But, — being equal to the constant quantity w, x= r cos 0, — - = cos — - - wr sin 0, dt df d^x f. dh' ^ . f. dr ., f, , — — = cos (7 —zr-o - 2m sin tf -rr- io'r cos a .... (2) ; df df dt ^ ^' and y = r sin 9, -4- = sin ^- + w/' cos H, ^ dt dt d'y . ,, f/^r „ .. dr . ■ /, / , —4 = sm u — — + 2w cos i) - — oj'r sm U (3). df df dt ■ Yrom (I), (2), (S), ^ve get d\ .. „ drr , df df 178 FREE CURVILINEAR MOTION OF A PARTICLE. multiplying by 2 — and integrating, dr'' dr . let a be the initial value of r ; then, since r = a, -— = 0, simul- dt taneously, C = - w'ar ; hence dr' dB or, since — - dt df ^^"-^^' dr' , o ,/i dr - a', dff = (r - aj dO' Integrating, we have + C = log [r + {r' - a-j] ; but B = 0, r = a, simultaneously, and therefore C = log a ; hence B = \ (r (r \n (4) A particle moves with a uniform velocity under the action of two centres of force, each varpng inversely as the distance, and of equal intensity at the same distance ; to find the path of the particle, its initial position coinciding with the middle point of the straight line which joins the two centi-es of force. Let the initial position (fig. 101) of the particle be taken as the origin of co-ordinates ; let C, C , be the two centres of force ; OCz the axis of x, Oy at right angles to Ox the axis of y ; 0C= OC ^a; 031= x, PM = y, OP = s, CP = r, C'P = r ; j3 = the constant velocity of the particle. Then, by hypothesis, ds' da? dy' „, — or V -^- - B de df df ^ Diflferentiating, we have dx (Tx dy d^ ^ df df '^ df df ^ ^' FREE CURVILINEAR MOTION OF A PAP.TICLE. 179 Again, jj. denoting the absolute force of attraction of each centre of force, we have, by Maclaui'in's Equations (A), cPx /J. a - X fji a + X df r r r' r df r r r' r' ' and therefore, from (1), dx fa + X a - x\^ dy fy y dt \ r'~ r^ / dt \r^ r {x + a) dx + ydy (x - a) dx + ydy (x + df + y'^ ' {x - af + y- Tntegrating, we have log [{x + af + y'] + log {(x - af + y'] = log C, [{x + af + 2/'} [{x - af + y'] = C. But X = 0, y = 0, simultaneously ; hence C = a^, and therefore {(x + af + y-} {(x - af + y'] = a\ (a' + x^ + y'^f - \d'x' = «*, 2a^ (x^ + y-) + (x- + y-f - 4aV = 0, (x' + tff = 2«- ix' - y'), which is the equation to the Lemniscata of James Bernoulli, i i'a being the semiaxis of the corresponding equilateral hy-, perbola. (5) A particle attracting with a force varying directly as the distance moves uniformly in a straight line; to determine the motion of another particle situated in the same plane and subject to its influence, the initial circumstances of the latter particle being given. Let the initial position of the attracting particle be taken as the origin of co-ordinates, and let the axes of x and y make angles of 45° on each side of the line of its motion. Let X , y , be the co-ordinates of the attracting, and x, y, of the attracted particle at any time t, then the equations of motion IX denoting the absolute force of attraction. n2 180 FREE CURVILINEAR MOTION OF A PARTICLE. But if a, (5, denote the resolved parts of the velocity of the attracting particle parallel to the axes of x, y, ^vhich are by hypothesis invariable, x = at, y' = ^t, and therefore ^2^ — = IJL-iat- x) (1), §^ = /r^^-2/) (2> From the equation (1) -p (a; - at) -i- fx' (x - at) = 0. The integral of this equation is X ^ A cos (fit) + ^ sin (/ut) + at (3) ; where A, B, are arbitrary constants. Differentiating we have -— = - An sin {fit) + Bji cos {fit) + a (4). at dx Let a, m, be the initial values of a:, -f-; then, from (3) and (4), at a = A, m = B[i -: a, and therefore, from (3), a; = a cos (jit) + sin {/it) + at. In precisely the same way, b and >i being the initial values of u and 7? /3 ^ dt ' y = J cos {fit) + '- sin {fit) + (3t. (6) A particle urged towards a plane by a force varying as the perpendicular distance from it, is projected at right angles to the plane fr-om a given point in it with a given velocity ; to determine the force which must act at the same time on the par- ticle parallel to the plane, that it may move in a given parabola having its axis in the plane, and to find the co-ordinates of the particle at any epoch of the motion in terms of the time. Let the initial place of the particle be taken as the origin of co-ordinates, the axis of the parabola as the axis of x, and a straight line at right angles to the plane through the origin as FREE CURVILINEAR MOTION OF A PARTICLE. 181 the axis of y. Then, since the required force must evidently act parallel to the axis of x, we have, by Maclaurin's Equations (A), d^x le-^' ^^^' ^'-'^y (^>' where X is the required force and fx a constant quantity. Also the equation to the parabola will be y^ = Amx (3). Differentiating this equation, chf ^ dx '' dt dt ^+y_4=2w-r-5- = 2mX, by (1), (4). df ^ dir df > j y ^> v ; Multiplying (2) by 2 -^ , and integrating, dt' ^^ Let V be the velocity of projection ; then, y being initially equal to zero, V^ = C, and therefore f=^=-r (^). Hence, from (2), (4), (5), 2//^X= V^-2fiy\ X = r= 4u:c, by (3) (6) which determines the required force. Again, from (5), df 1 I } Integrating, we have ^=C7+lsin-^: 182 FREE CURVILINEAR MOTION OF A PARTICLE. but y = when t = i), and therefore C= ; hence /=lsin->^, y=^^(^}f), ....(7). Also, from (1) and (6), dx cT'x V^ dx dx 2 -y- — r = r- - 8ju.^ ^ . e*"' ?/ sin a - u. sin a, e^~^ (6). From (5J and (6) we get tan a^ = tan a^ . e*'^, u^ = u^ cos a, . {l + tan'a^ . 6^"""'^}% by which the circumstances of the projection are determined for each of the parabolic paths. 184 FREE CURVILINEAR MOTION OF A PARTICLE. Again, from (3) and (6), 2 . .^. A^ = ~ u. sm a, e' {7); ' ff integrating and adding a constant, 2u, sin a, e' ^ t^ = —l -+ C; 9 e-\ but ^0 = Oj hence ^^2j^^^ina, _±_^(j^ 9 e-\ -. . r ^ 2w, sin a, e^ - 1 and tnereiore t = — . 9 «- 1 Again, £i-om (4) and (7), 2 A* = - u, sin a, . e' ■ u , cos a., and therefore, by (5), M," sin 2a. _ As = -^ -^ e^, 9 whence, integrating and observing that s^= 0, we shall have u^ sin 2a, \ - e' 9 1-^ Bordoni; Memorie della Societa Italiana, torn. xvii. P. 1, p. 191 ; 1816. (8) A particle is projected obliquely from a point A at an angle a with the horizon, so as to hit a point B, AB being inclined at an angle )3 to the horizon ; and the velocity of pro- jection is such that with it the particle would describe the straight line AB uniformly in ti seconds ; to find the time of flight. ^D Required time = n cos /3 cos a (9) If two particles be projected from the same point, at the same instant, with velocities v and v , and in directions a, a', to find the time which elapses between their transits thi'ough the other point which is common to both their paths. rp. _ 2 r^-' sin (a ..^ a) g c cos a -\- c cos a FREE CURVILINEAR MOTION OF A PARTICLE. 185 (10) If a be the angle between the two tangents at the ex- tremities of any arc of the parabolic path of a particle acted on by gravity ; v, v , the velocities at these two points, and v^ the velocity at the vertex ; to find the time through the arc. rp, . 1 . . vv sin a ihe required time = . (11) A particle describes an ellipse under the action of a force at right angles to the axis major ; to find the force at any point of the path. Let a, h, be the semiaxes major and minor, y the distance of the particle at any point of its path from the axis major, j3 the velocity of the particle parallel to the axis major which will remain invariable during the whole motion. Then force required = -^ . ay \i h = a, or the ellipse become a chcle, force = —V . y Riccati; Comment. Bonon. toxa. w. ^. 149; 1757. New- ton ; Principia, lib. i. sect. 2, prop. 8. (12) A particle describes the arc of a cycloid under the action of a force parallel to its base ; to find the law of the force. If the equations to the cycloid be X = a vers B, y = a (6 + sin 6), and F, /3, denote the force required and the velocity parallel to the axis of the cycloid, l^^J^ (2sin0-sin 29). (13) A particle is projected with a given velocity parallel to a given straight line towards which it is always attracted with a force proportional to its perpendicular distance from it; to determine the position of the particle at any time and the equa- tion to its path. Let A (fig. 102) be the initial position of the particle ; Ox the given straight line, yAO at right angles to Ox; Ox, Oy, the 186 FREE CURVILINEAR MOTION OF A PARTICLE. axes of X, y; P the position of the particle after a time t ; OM = X, PM= y; AO = h, /3 = the velocity of projection ; /x* the absolute force of attraction. Then iiX X = ^t, h cos —-= y = I) cos (fit). Riccati; Commetit Bonon. torn. iv. p. 155; 1757. (14) A particle is projected from a point x = 0, y = b, with a velocity j3 parallel to the axis of x, and is subject to the action of a force tending towards the axis of x parallel to the axis of y, and varying inversely as the square of the distance ; to find the equation to the path of the particle. Let fx denote the attracting force at a unit of distance ; then the equation to the path will be Riccati ; lb. p. 159. (15) A particle (fig. 103) is projected with a given velocity in the direction Oy, and is acted on by a centre of force, which attracts dii-ectly as the distance and moves uniformly with a given velocity along Ox at right angles to Oy ; to determine the position of the particle when its motion first becomes parallel to Ox. Let fx^ denote the absolute force ; a the initial distance of the centre of force from O ; /3 the velocity with which the particle is projected, j3' the uniform velocity of the centre offeree along Ox, and x', y , the co-ordinates of the required position of the particle ; then ^ = « + ^ C'T - 2), y ='- . ri6) A particle which is placed at rest initially in a given position, is acted on by two forces, one repulsive and varying as the distance from a given point, and the other constant and acting in parallel lines ; to determine the position of the particle at any time and the equation to its path. FREE CURVILTNEAK MOTION OF A PARTICLE. 187 Let the centre of the central force be taken as the origin of co-ordinates, and let the directions of the axes be so chosen that the direction of the constant force makes an angle of 45° with each of them. Then if a, b, be the co-ordinates of the initial position of the particle, ^i" the absolute force of repulsion, and /' the constant force, we shall have, putting y= 2"V"''^j = h U^' + e ''') = '} . a + m ~ b + m (17) Four equal particles attracting directly as the distance are fixed in the corners of a square ; to find the path of a par- ticle projected from any point and in any direction in the plane of the square, Let the centre of the square be taken as the origin of co-ordi- nates, and let the axes be at right angles to the two pairs of opposite sides of the square. Then if a, b, be the co-ordinates of the initial position of the particle, 2m, 2n, the resolved parts of its velocity of projection parallel to the axes of x, y, respec- tively, and \jc the absolute force of attraction of each of the fixed particles, the equation to the path of the free particle will be .,[1X1 . , uh . , \xx . fia (18) A spherical particle, of which e is the elasticity, is pro- jected with a velocity v at an angle of elevation a, and at the instant of attaining its greatest altitude strikes a similar and equal particle falling downwards with a velocity ^v at the point of collision ; to find the distance of the particles from each other at the end of t seconds after the impact. J. Distance required = let (1 + 4e^ cos' of. (19) A heavy particle, having been projected at a given angle to the inclined plane AJB, (fig. 104), proceeds to ascend this plane by bounding in a series of parabolic arcs ; to determine the angles of incidence and reflection after any number of impacts. 188 FREE CURVILINEAR MOTION OF A PARTICLE. Let I be the inclination of the phme AB to the horizon ; a^ the angle of reflection in the x^^ arc, /3_^_j the angle of incidence ill the (x - 1)* ; and e the elasticity of the particle. Then (1 - e) ef'^ tan a, ^ o tan a = — , - ^ — .- - ' = e tan p ,. * 1 - e - 2 (1 - e'-') tan t tan a^ '^""' Bordoni ; Memorie della Societa Italiana, torn. xvii. P. l,p. 191; 1816. (20) A baU, of which the elasticity is e, is projected with a velocity F" in a dii-ection making an angle a + t with the horizon, and rebounds from a plane inclined to the horizon at an angle i and passing through the point of projection To determine the relation between R^, H^^^, ^^2> ^^'^e con- secutive ranges upon the inclined plane after x, x + l, x+2, rebounds respectively, and to find the sum of all the ranges on the inclined plane before the ball begins to slide down the plane. If cot j3 = ( 1 - e) cot I, and S denote the sum of the ranges ; B^^^ - (e + e') R^^^ + e'B^ = 0, ~ _ 2 F^ sin /3 sin a cos (a + j3) g sin t . cos^ /3 Sect. 2. Central Forces. Let the force which acts on a particle tend always towards a fixed centre, which we will take as the origin of co-ordinates. Let F denote the force at any distance from the centre ; x, y, the co-ordinates of the particle at the end of a time t reckoned from an assigned epoch, r its distance from the centre of force, and 9 the inclination of this distance to any fixed line in the plane of X, y. Then, by Maclaurin's Equations, the plane of co-ordinates being identical with the plane of the motion, cPx _ Fx d~y Fy 'df~~~7' 'df^~V From these equations may be obtained the following formulae : rdB^hdt (I), v = - (11), P FREE CVRVILINEAR MOTION OF A PARTICLE. 189 '^^'^m^^ ™- V = V 2 Fdr (IV), ^=11 (^^)' i.=.^^^ (VII). df df ^ ^ In these formulae h represents twice the area swept out by the radius vector about the centre of force in a unit of tinie,^^ the perpendicular from the centre upon the tangent at any point of the orbit, v the velocity of the particle, and v , r , any simulta- neous values of v, r. If the central force, instead of being- attractive as we have been supposing, be repulsive, we must replace F in these formulae by - F. The equation (I) shews that the area swept out by the radius vector varies as the time, and either of the equations (II) and (III) that the velocity at any point of the orbit varies inversely as the perpendicular fi'om the centre of force upon the tangent to the orbit at that point : these two propositions were first established by Newton\ The equation (IV) shews that the velocity of the particle at any point of its path depends only upon the distance of the point from the centre, the velocity of projection and the prime radius vector, whatever be the course which it may have pursued ; the discovery of this proposition is likewise due to Newton^. The formula (V), by which the path of the particle may be determined when we know the law of the central force and conversely. Ampere^ ascribes to Binet. The formula (VI) was communicated without demonstration to John Bernoulli by De Moivre in the year 1705; a proof of the formula was returned to him by Bernoulli in a letter dated ' Princ/pia, lib. I. Prop. 1. ' lb. lib. I. Prop. 40. ' Annates de Gergonne, torn. xx. p. 5.">. 190 FREE CURVILINEAK MOTION OF A PARTICLE. Basle, Feb. 1706. The formula (VII) was given much about the same time by Clairaut^ and by Euler', and signifies that the acceleration of the radius vector is equal to the excess of the centrifugal above the attractive force. (1) To find the law of the force by which a particle may be made to describe the Lemniscata of James Bernoulli, the centre of force coinciding with the node, and to investigate the time of describing one of the ovals. The polar equation to the Lemniscata is r" = «- cos 20 (1); , 1 1 d fW sin 2d hence '" « (cos 20/ "^^^''^ a (cos 20/ d- {\\ 2 3 (sin 20)' 3 1_ a (cos 20)* a (cos 20)* a (cos 20)" a (cos 20/ and therefore (P_ / 1 \ 1 3^ _ 3a* dO' \r) " r ~ ^, „T4 ~ 'V ■ ^ ^ a (cos 20)- Hence, by the formula (Y), ;•' Again, by the formula (I) and the equation (1), we have /idt = rdO = a' cos 20 dO, and therefore, if P denote the required periodic time, -r. a' f'l'^ ^ ,„ cr P = — cos 20 f70 = ^ . h J -iTT /' Let n denote the value of F when r = \ ; then we have ^=3«v,^ h = -^, p = ^. 3V )u^ (2) A particle moves in an equiangular spiral under the action of a force tending towards the pole ; to find the law of force and the velocity at any point of the orbit. ' Theorle de la Lune, p. 2 ; the first edition of which appeared iti 1752, from a MS. sent to St. Petersburg in 1750. " Nov. Comment. Petrop. 1752, 1753, p. 164. FREE CFKVILINEAR MOTION OF A PARTICLE. 191 If j3 be the invariable angle, r the radius vector, and p the perpendicular from the pole upon the tangent, p = r sin j3 (1). DiiFerentiating with respect to r, we have -j- = sin j3, and therefore, from (VI), ^=^sin/3=-^4V^, by (1), . . . . (2). p r sin p Let c be the velocity corresponding to a given radius vector r ; then, by (II) and (l), h = cr sin /3. Hence, from (2), F = -^ , and, from (II) and (1), cr' sin i3 cr V = ,—^ = — . r sm p r (3) A particle describes an equilateral hyperbola round a centre of force situated in the centre ; to find the law of the force and the angle which the particle will describe about the centre from the apse in a given time. The equation to the hyperbola being 'lY_ cos 20 rj Cl- in have iin^-^-^^-^ (2), r ciO \r J a- 1 cf- r cW ©• \dO G)'!- 2 cos 20 and therefore, from (2), r dO' ' ?)• r' . -- sm a ;- 20 = - 2 cos 20 and thence. by (1), 1 cP 1 r dO' [ })■ r- 1 a' r' 2 r cr cW 0.) 1 r r' a' COS 20 192 FREE CURVILINEAR MOTION OF A PARTICLE. Hence, bv the formula (Y), F-'lL. a The negative sign shews that the force must be repulsive ; let - fjt be the absolute force, that is, the value of i^ at a unit of distance. Then 73 M = -4 ' ^ = - ."'■• a - cr Putting for h its value ff'/i" i^ the formula (V), and for r^, we get de i,^ cos 26(19 i f7sin20 ^ i , ^Tf.^ u'at, r— ^ — ^ = u'f, ^^ — _ = 2u7/r; cos 20 ^ 1 - snr 29 \ - snr 29 integrating, and supposing the time to be reckoned from apsidal passage, we have , , 1 + sin 20 „ i 5 log . ^^= 2,1 1, 1 - sm 29 i whence, writing /x' in place of 4^', we obtain sin 20 = '"^ . £"''+1 (4) A particle is revolving in a parabola about a centre of force in the focus, and when it arrives at a given distance from the focus, the absolute force is suddenly doubled ; to determine the natui-e of the subsequent path of the particle. Let 4?« be the latus rectum of the parabola, r the radius vector at any point, and p the perpendicular from the focus upon the tangent. Then, by the nature of the parabola, 1 1 2 dp _ \ jr mr ' // dr mr ' and therefore, by (VI), ^= 5- 27nr But, after the absolute force has been doubled, we shall have for the motion 7- mr FREE CURVILINEAR :\IOTION OF A PARTICLE. 193 and therefore, by (VI), W f dr ' 1 1 ? dp dr Integrating, we have 1 inr = C 1 Let c be the value of r at the instant when the absolute force is doubled ; then, ^j being then common both to the parabola and to the new path, we have mc 2mc 2mc and therefore, for the equation to the new path, there is 111 mc 2c mr 2mc 2p' ^ p" r which is the equation to an ellipse, 2c being the major and {jncf the minor axis. Since the elKpse touches the parabola when r = c, the semi- axis major, it follows from the nature of the ellipse that the point of contact is an extremity of the semi-axis minor, and therefore that the axis major of the ellipse is parallel to the tangent at the point r = c of the parabola. But the sine of the angle of inclina- tion of the tangent of the parabola at this point to its axis is ^ = — , when r = c, that is, = — ^ > c^ of the major axis of the ellipse to the axis of the parabola is when r = c, that is, = — , and therefore the inclination ^ r^ c^ (5) A particle is describing a curve about a certain centre of force, the velocity of the particle varying inversely as the w"' power of its distance from the centre of force ; to find the law of the force and the equation to the path. We shall have, /ul denoting some constant quantity, 194 FREE CURVILINEAR MOTION OF A PARTICLE. HeucGj from (IV), there is and therefore, differentiating, ^^^ Avhich determines the law of the force. Again, from (III) there is cie r r'f /r/-'"-='V ; d 1 11 -T-„ ,.2« (71- i)de dS r dr' (w-l)0 = C+cos-^ -— -. Suppose 0=0, when r = a; then, k denoting a constant quantity, (n- 1)0 = cos-' (Av-"-') - cos-' (ka"-'). Riccati; Comment. Botio7i. torn. iv. p. 184. (6) If the force vary as the 7i^^ power of the distance, and a particle be projected from an apsidal distance, with a velocity of which the square is equal to 1 - £ times the square of the velocity in a circle about the same centre of force with a radius equal to the apsidal distance ; to find the equation to the orbit, £ being a very small quantity. Let a be the apsidal distance ; then r = a - x, where a; is a small quantity, because the path of the particle, as is evident from the initial circumstances, will be nearly circular. Then, approx- imately, 1 1 1 /^i ^^ d' fl\ 1 d' r a - X a\ a J dO' \r J a' dff^ Also, ju denoting the absolute force of attraction, Fr' _ f.r"' _fi _ ^a-^ f (n + 2)x } FREE CURVILINEAR MOTIOX OF -* PARTICLE. 195 Hence, by the formula (V), 1 d~x X 1 fxa"*' C (71 -\- 2)":^'] _ + |W(„+2)M_|, + „__=0 (1). Let V be the velocity of projection, and v the velocity in a circle about the same centre of force with a radius a ; then F' = (1 - i)v' = (!-£) fxa^*' (2). But, by the formula (II), because the motion is initially at right angles to the radius vector, and a, V, are the initial values of the radius vector and of the velocity. Hence, from (2), h^ = ;u (1 - e) a" ' h^ (1 -E)a"^ and therefore, from (1), the product of e and x being neglected, -—7 + (« + 3) a: - m = 0. clu Multiplying by 2 -^ and integrating, du ^3 + (w-f 3).r-2£«.r = 0, no constant being added because -^ is by hypothesis equal to du zero when x = Q ; hence dx de (w + 3)' ( z - .rM- ^ ^ \n + 3 / sidering zeri (n + Sf = vers" Integrating and considering zero to be the initial value of 9, J (w + S)x whence for the polar equation to the curve we have r = a- -^ vers {(ji +3/0} (3). w + 3 ^ ' o2 196 FREE CURVILINEAR MOTION OF A PARTICLE. The general condition for an apsidal distance is evidently that -j^ shall be equal to zero : differentiating the equation (3), we get for the determination of apsides, ^ = -, sm {(n + sy Q] = 0. {n + 3/ i_ Hence {n + 3)' Q = Xtt, where X is any integer : let ff, B' , be the values of d for two con- secutive apsidal distances; then {n + 3)^ & = Xtt, (n + sf &' = (X + 1) ir, and therefore,

dy d-y 1 dv 1 dy dx doi? V ds p dx ds^ ' dx' dy d\j dy d'y 1 dv dx dx' dx dx' V dx ds' dy' dx' ^ dx"^ Integrating, we get log v = C+\ log (\ + ^^) = C+ log \ ClX J ds dx Let V denote the initial velocity, and j3 the angle which the direction of projection makes with the axis of x ; then log V = C -^ log sec /3, ds^ V d'T and therefor~e loof — - = log . " , , ° F '^ sec j3 V = V cos Q -r- . dx Substituting this value of v in (2), we have y^ V ops' j3 d^ p dx^' Euler; Mechan. torn. i. p. 240. (3) A particle describes a given curve about a centre of force ; to determine the motion of the particle and the law of the force. Let APB (fig. 105) be the path of the particle, S the centre of force; P the position of the particle at any time; PT a. tangent at the point P, and SY perpendicular to PT. Let F FREE CURVILINEAR MOTION OF A PARTICLE. 205 denote the force along PS, and

(1;, - = jP sin (2). P Now, since ds cos = dr, vdv , dv and p sin d> = —- sin ^ = P -r- > dp dp where j5 denotes SY, we have, by (1) and (2), vdv = - Fdr (3), ^'= Fp% (4). dp Eliminating F between (3) and (4), dv dp , ^1 — ■ = — — , log «? = (;- log p. V p Let V, P, be the initial values of v, p ; then log F= C-logP, V P V.P and therefore log -— ; = log — , v = — '- — (5). V p p Again, if t denote the time of the motion, ds V.P , .,. ~ = v= _ , by (5), dt p pds = VPdt; hwi pds is equal to dh' , where h' represents twice the area swept out by the radius vector in its motion from some assigned posi- tion ; hence ^j, ^ ^p ^^^ ^, ^ yp^ (g)^ the area being supposed to commence with the time. Again, by (2), we have ^ v^ v^ dp V-P~ dp , ... f^. F= — ^— = - -f = — — -f ' by (5), (7). p sui(p p dr p dr Suppose now that h represents twice the area swept out in a 206 FREE CURVILINEAR MOTION OF A PARTICI-E. unit of time ; then, since by (6) A is equal to VP, we have b)' (6), (■^^'Ol h' = kt (8), ^J ^^)' F=^f^^ (10). The formulae (8) and (9) were given by Newton.^ Formula (10) was discovered by De Moivre in the year 1705, by whom it was communicated -^-ithout demonstration to John Bernoulli, A proof of the formula was obtained by Bernoulli and forwai'ded to De Moivre in a letter dated Basle, Feb. 16, 1706. Demon- strations were afterwards given by Keill," and by Hermann.^ See De Moivre's Miscell. Analyt. lib. viii., and John Bernoulli, Opera, tom. i. p. 477. Integrating the equation (3) we get another expression for the velocity rr v^= V'-2 Fdr, J R where M denotes the prime radius vector. This result shews that the velocity of the particle depends only upon its distance from the centre of force, and not upon the path described ; a theorem proved by Newton.* Euler; Median, tom. i. p. 240. Sect. 4. Motion in Resisting Media. (1) A particle acted on by gravity is projected in a uniform medium, of which the resistance varies as the velocity, with a given velocity and at a given angle of inclination to the horizon ; to find after what interval of time the particle will arrive at its greatest altitude. Let k be the resistance for a unit of velocity, u the velocity and a the angle of projection, and let y be the height through ' Principia, lib. i. Prop. 1. - Phil. Trans. Num. 317; 1708. ' Phorunomia, p. 70. * Princip. lib. i. Prop. 40. FREE CURVILINEAR MOTION OF A PARTICLE. 207 which the particle has ascended at the end of the time t. Then d'^y , ds dy , dy but -r- = u sin a when ^ = ; hence dt and therefore log {g + ku sin a) = C, , q + ku sin a , , log li = kt. ^ dt When y is a maximum, -^ = 0, and therefore the required value of t will be equal to - log (1 + - w sin a). ^ 9 (2) A particle moving in a resisting medium is acted on by a given force in parallel lines ; to find the resistance that any pro- posed curve may be described, and conversely. Let the positions of the particle be referred to two rectangular axes Ox, Oy, (fig. 106), OM = x, PM=y, AP ^ s ; P being the position of the particle at any time, and APB the cur^^e of its motion ; also let Y denote the accelerating force at P parallel to Oy, V the velocity of the particle, and H the resistance of the medium. Then, by the equations of tangential and normal resolution given in the preceding section, we have ,^4 = -R^Y^ (1), ds ds t^r^^ (2); p as where p denotes the radius of curvature at P. But d^ dx^ dx^ 208 FREE CURVILINEAR MOTION OF A PARTICI.E. hence, from (2), ds^ differ eutiating, we have, since y-^ = l + ^ ^ , dx'^ d^^^^dl dx^ dx^ dv -^ydxi 1 ds' d V — = Y-^ + dx dx 2 ds^ dx dv ^^dy 1 ds d V — = y— +— — ■ — ds ds 2 dx dx hence, from (1), B = — — — - \-Tr- ^ 2 dx dxi d-y vdx' . which gives the resistance if the curve be given, and conversely. The solution of this problem, which Newton had given in the first edition of the Principia, involved certain errors, which at the suggestion of John Bernoulli were afterwards corrected. CoR. If the resistance vary as the square of the velocity for a uniform density ; then, Q denoting the density generally. we have R = Q>f = Qp '^ Y, by (2), d^ R dx' and therefore ^ = tt 2 1 dx dx' d ( Y ] ~~ 2 'ds T dx \^\ .-L^^logfZ.) 2 ds dx '' wr idx?) which gives the density at any point within the medium ; or, if the density be given, determines the curve. FREE CURVILINEAR MOTION OF A PARTICLE. 209 Cor. It is evident that p — is equal to half the chord of ciu'vature at P parallel to Oy, or in the direction of the force Y; let q denote this chord of curvature. Then v" =1Y .\q; and therefore the space due to the velocity, supposing the force to continue constant, is equal to one-fourth of the chord of curvature. Newton; Princip. lib. ii. prop. 10. John Bernoulli ; Act. Erudii. Lips. 1713; Opera, torn. i. p. 514. (3) A particle moves in a resisting medium under the action of a given force always tending towards a fixed centre ; to determine the law of resistance when the path of the particle is given, and conversely. Let APB (fig. 107) be the path of the particle, S \he centre of force, AP = s, SP = r, p= the perpendicular from S upon the tangent at P, v = the velocity at P, p the radius of curvature, P the central force, and P the resistance of the medium. Then, by the equations of tangential and normal resolution, ^^ ^^^^ V — --R-P- (1) ds ds t^Pp (2). p r dv Since p is equal to r ^- , we have, from (2), dp "'-''i^ w^ and therefore, difierentiating with respect to s, dv \ d f dt -p ds 2 ds\ dp 2 ds^X dp Jp'j p' ds Y dp J 2/ ds V dp J dr 2jo^ ds y dp , P+— , Z.{j/:^p ds . . . . p 210 FREE CURVILINEAR MOTION OF A PARTICLE. Hence, by substituting this value of v — in (1), we have ^-^'iVi") «' which determines the law of resistance when the curve is known, and conversely. Cor. Supposing the resistance to vary as the density of the medium multiplied by the square of the velocity of the particle, we have, Q denoting the density. and therefore, by (4), It = Qv'=Qp'^P, by (3), d f d^ 1 dsVdp) _ _ 1 d__. / 3 dT_ dp which determines the law of the density when the curve is given, and conversely. Cor. From the equation (2) we have v'^^-^pP = lqP=2{\q)P, where q denotes the chord of curvatm'e through ^S*. This result shews that the velocity at any point of the cuiwe is that due to falling in vacuo towards the centre of force, continued constant, through a quarter of the chord of curvature. CoR. From (4) we have dp p where h is some constant quantity. This formula gives the central force when the law of the density and the orbit are given. It is easily shewn that, if ASP = Q, !^±_h' fl d^ fl\\ y dr ~r' {r'^ dd'\rjj' FREE CURVILINEAR MOTION OF A PARTICLE. 211 Und therefore P = ^' (1 -f 4J^ f^)] e-2/«''». r" [r (W~ \rjj If Q = 0, we get P = ~ {- + -jt^A ~]] ^ ^ r {r dd- \rj\ which is Binet's formula for the central foTce in vacuo. Newton; Princijna, lib. ii. Prop. 17, 18; John Ber- noulli; Opera, torn. iv. p. 347. Euler; Median. torn. I. p. 428 et sq., p. 451 et sq. (4) A particle is projected with a given velocity in a uniform medium, in which the resistance varies as the square of the velocity ; the particle is acted on by gravity, and the direction of its projection makes a very small angle with the horizon ; to determine approximately the equation to the portion of the jjath which lies above the horizontal plane passing thi'ough the point of projection. Let Ox and Oy (fig. 108) be the horizontal and vertical axes of X and y, being the point of projection; P the position of the particle at any time; OM=x, PM = y, v = t}ie velocity at P, s = the arc OP, k = the resistance for a unit of velocity ; then, by the tangential and normal resolutions, .^^-to'-y-^ (1), as as 2 dx I +p^ ^„. "'^^''--''-T' ^^' where p = -^ and Q = -~ • Hence, eliminating v between these dx dx two equations, we have ds q q ds f. m + ul^f-ip^o, dx q q dx d , 1 + «^ r.1 ds 2pq — log i- ^ 2k- ^ = 0. dx q .dx 1 +p' Integrating, we get 1 + ?/ log i- + 2/i-s + log C - log (1 +^/) = 0, P 2 212 FREE CURVILINEAK MOTION OF A PARTICLE. C being some constant quantity ; log -^ 2hs = 0, q= Ce'"' (3). 2 Let w be the velocity and a the angle of projection; then, initially, by (3) and (2), ^ , 1 + tan^ a and therefore hence by (3), but, the angle of projection being small, we may neglect all powers of jo beyond the first, and therefore 5 = I (1 +p'y dx = \ dx = x nearly; Jo Jo hence " / 9. c 9 u' cos 'a' ^7" 9 cos ~a gZks. p= C- „2kx . ikv^ cos^ a but p = tan a when .r = ; hence tan a= C ^ llit^ cos^ a and therefore p = tan a + ^ ^ — r— (1 - e^^\ 2ki( cos a Integrating again, y ^ C + X tan a + -r~A — ?— ( ^ f «"' ^ 2^w' cos' a V 2^ but ;-" ; then i? = -^(o + «)?-sin6l, Q = -(w + 5)—. 4 ^ 2 ^ r FREE CURVILINEAR MOTION OF A PARTICLE. 215 (9) A particle moves in an equiangular spiral about a centre of force in the pole whicli varies as any power of the distance from the pole ; to find the law of the resistance and of the den- sity of the medium, the resistance being considered equal to the product of the density and the square of the velocity. If a be the constant angle, r the radius vector at any point, lj.r" the attractive central force, and the particle be so moving as to approach the centre of force ; -r^l/ N ^1^ N cos a M = - fx{7i -{■ 3) r" cos a, Q= -{n + 3) — — . Newton; Princip. lib. ii. Prop. 15, 16. John Bernoulli; Opera, tom. iv. p. 350. (10) A particle moves in the circumference of a circle about a centre of force in the centre ; the resistance of the medium in which the motion takes place is constant ; to determine the law of the force, the velocity at any time of the motion, and the time which elapses, as well as the space which is described, before the particle is reduced to rest. Let /3 be the initial velocity of the particle, a the radius of the circle, c the constant value of the resistance, s the arc described from the beginning of the motion, P the central force ; then v' = ^'-2cs, P = -(ff -2cs); a when the particle is reduced to rest, 2c c (11) A particle is moving along the curve of an equiangular spiral so as to be approachmg the pole ; the motion takes place in a medium where the resistance varies as any power of the distance from the pole ; to find the law of the central attractive force in the pole. Let a be the constant angle, j3 the initial velocity, a the initial distance, h^ the resistance at a distance r, P the requii'ed force; ^^^^^^ p _ (n + 3) cos a a'[5- + 2k (?-"^^ - a"^) {n + 3) cos a r^ Euler : Mechrni. tom. i. ]i. 442. 216 FREE CURVILINEAR MOTION OF A PARTICLE. (12) A particle is projected with a velocity u, and at an inclination a to the horizon, in a uniform medium where the resistance varies as the velocity; to determine the time which elapses before the direction of the motion is inclined to the horizon at an angle /3. If k represent the resistance for a unit of velocity, t will be found from the equation g cos /3 (i*' - \) = ku sin (a - /3). (13) Two j)articles, subject to the action of gi-avity, are simul- taneously projected at equal angles of inclination to the horizon, and with equal velocities, the one in vacuo and the other in a medium where the resistance varies as the velocity ; to deter- mine a relation between the times in which the particles describe two arcs so related to each other that the tangents at their ex- tremities shall make equal angles ■ndth the horizon. If k denote the resistance of the medium for a unit of velocity, and t^, t^, denote corresponding times in vacuo and in the medium ; then (}-h = j + ^^ . (14) Having given the co-ordinates of the highest point of the path described by a particle acted on by gravity and pro- jected in vacuo at a known angle of inclination to the horizon; to find the decrements of these co-ordinates when the particle is projected m a rare medium in wliich the resistance varies as the velocity. Let a, b, be the given co-ordinates, k the resistance for a unit of velocity, and /3 the angle of projection; then Sect. 5. Impossible Mechanics. The determination of the motion of a material particle in a plane XOY, (fig. 112), under the action of assigned forces in that plane, depends upon the integration of two simiJtaneous difierential equations ^-*(.r,y), J'x{-,y) (1), FREE CURVILINEAR MOTION OF A PARTICLE. 211 where x, y, are the co-ordinates of the position of the particle, at the end of any time t, referred, we will suppose, to rectangular axes OX, Y, and ^ {x, y), ^ (z, y), are certain functions of X, y, depending upon the nature of the forces. Suppose that, for certain particular forms of the functions, having effected the integrations and determined tlie arhitrary constants from the initial cuxumstances of the motion, we are enabled to obtain the relations ^-0,(0, V - X.H) (2), where (f>^ (t), ■^^ (f), represent certain functions of f. In the determination of these relations is generally supposed to consist the complete solution of the problem of the motion of the par- ticle. There are particular cases, however, in wldch they cease, after the lapse of a certain time, to correspond to the physical motion of the particle. It occasionally happens that as t increases indefinitely from zero by continuous gradation, one or soraetimes both of the functions ^^ (t), ^^ (t), after remaming possible up to a certain magnitude of t, at length assumes impossible values. Before proceeding to the consideration of the mechanical circumstances connected ^\dth this anomaly, it will be necessary to make a few observations on the geometrical signification of impossible values of x and y. By the theory of impossible quantities we know that for all values of x and y, possible or impossible, we may assume X = (cos B + -' sin 0) a, y = (cos ^ + - * sin ^) /3, where a, j3, 6, 0, are all possible and positive quantities. Sup- pose now that B = iXir, where X is an integer ; then we have X = a; also, if = (2 A + 1) tt, where A is still supposed to be integral, then x = - a. It is evident then that whatever geo- metrical interpretation may be given to the general form of the value of x, it must be such that, when B is any even midtiple of TT, X may denote a distance a estimated along the axis OX. in the positive direction ; and that, when B is any odd multiple of tt, X may denote a distance a estimated negatively from 0, that is, in the direction XO. Similar remarks, mutatis mutandis, are evi- dently applicable to the interpretation of the values of y. These 218 FREE CURVILINEAR MOTION OF A TARTICLE. are the only conditions to which we are subject in our selection of a geometrical interpretation of impossible values of the variables. From what has been said then, it appears that the conventional significations of the signs + and -, leave that of the more general sign cos + -' sin ^ in a great measvu'e arbitrary. In fact - X = (cos + -* sin 0) a may be taken to represent any line OA of a length u drawn from at an inclination B to the direction OX ; thus the locus of the extremity A will be a circle described with a radius a sin 9 about a centre in OX, at a distance a cos B from 0, the plane of the circle being at right angles to OX. In like manner y = (cos -f -' sin ,fx,v)=0 (3). Also, the effective accelerating forces on the point P parallel to OX, OY, must (when z and y become impossible) continue to coincide with the physical law of the forces. Hence we shall have two more requisite relations, G (a, j3, e, 0, fx, r) = 0, H(a, ^, B, , fx, v) = . . . . (4). Now a, (3, B, (j), are each known functions of t; hence, betAveen the relation (3) and the two relations (4), we may eliminate //, v, and thus we shall get an equation involving t and known quantities. Since, then, t is restricted to particular values, it is evident that the equations (2) cannot, for any law of constructing the axes, be enabled to represent the physical motion of the particle for impossible values of the variables. Whenever, then, x or y assumes an impossible value for any value of t, we must conclude that the physical motion of the particle cannot be represented by any single pair of equations (2). It will be necessary, on arriving at such a critical value of t, again to revert to the equations (1), to integrate them anew, and to determine the values of the arbitrary constants in ac- (Ix dti cordance with the values which x,y, — , -j- , have acquired at the conclusion of the preceding stage of the motion. It will be necessary to repeat this process whenever t makes x or y im- possible. Corresponding to each integration there will be a new equation in x and y, resulting from the elimination of t between the corresponding pair of equations (2). A portion only of each of the curves belonging to the equations in x and y „3 ^ ,7^ ~ ,.3 > vl> 220 FREE CURVILINEAR MOTION OF A PARTICLE. vnll be described l>y the particle, which will therefore pursue a path consistiiisr of fragments of a series of distmct curves. (1) A particle placed iii the plane XOY is attracted towards the axes OX and OY hj forces varying inversely as the cubes of the distances ; to determine the motion of tlie joai-ticle. This problem gives rise to the differential equations d'x m^ dry w* de^ "^ ' df^~/ Integrating these equations, we have dz^ . m*^ dir „ n^ df x' df If dx dy A, B, bemg arbitrary constants. Now, initially, -j- and ~ are Ctv Ctv both equal to zero, and therefore, if a, b, denote the initial values of x, y, we have A ni^ ^ -r. ^^* a' o~ Hence ^ = nA^ -K],'^ = n^(K -]-),... .(2). df \x' ay' dt~ Vy lyj' ^ ■' Now, since the particle will move towards Y, — will be dt negative, and thus dx , (a* - x'f axdx , ,^ — - = - in~ ; = - mdt ; dt ax . ^ 2 i {or - xy i and therefore, A - a («^ - x^f = - m^ t, where ^' is an arbitrary constant ; but x = a when t = 0, and therefore A' = Q : hence a (a^ - xy = t}iH X = a' 5- : 1 „;: (3). similarly y' = b^ — — • J As soon as t becomes greater than -^ , it is evident, from the former of the equations (3), that x becomes unpossible ; and. FREE CURVILINEAR MOTION OF A PARTICLE. 221 when t becomes greater than -5 , it appears in like manner from the latter that y becomes impossible. We will commence with the consideration of the motion parallel to the axis of x. When X becomes impossible, the former of the equations (3) can no longer represent the physical motion of the particle parallel to OX, and we must revert to the equation cFx m* where x is now to be estimated in the negative direction. Integrating, and adding an arbitrary constant A! , we have dx' ., m* ... :^ =^ +— (4). at X But in the beginning of the second stage of the motion, and at the end of the first, the velocity is the same. Hence, by (2) and (4), and therefore, from (4), d^_ J}__\_ df ~ "^ Va;^ d Extracting the square root, we obtain m'dt = + ^ {a' - x'J „ ,, axdx or mrdt = ^ , . (a' - xy Integrating these two equations, mH+ C, = -a K - ^y> i 7nH + C^ = + a{a~ - x^f. The former of these equations corresponds to the motion from the axis of y, and the latter to the motion which afterwards takes place towards it. To determine C, , we have x=0 when t = —, which ffives C, = - 2a' ; and thus, for the first part of the motion, mH -id = -a {d - x^. 222 FREE CURVILINEAR MOTION OF A PARTICLE. la' When t = — 5- J then x = «, and the motion of return begins in' which is defined by the second of the equations ; and C^ = - 2a^ Thus both integrals are comprehended by the equation a" (a' - x') = (mH - 2aJ. 3(c' After a time — - it is evident from this equation that x m becomes impossible, and then the third period of the motion of the particle takes place. It is evident that the particle will again cross the axis of y, and perform on the other side of it, parallel to the axis of x, a motion exactly similar to that of the second stage of the motion. Similarly for the fourth, fifth, &c. stages of the motion parallel to the axis of x. From the preceding conclusions it is manifest that, dividing the time from its zero value into a series of intervals of which the (^ , 2(C first is equal to — 7, and all the following ones to — - , the motion m' m' of the particle at right angles to OY will be represented for each interval respectively by the following equations, a" [a^ - (+ xj] = m'f, d {d' - (- xj] ={rnH - 2dJ, d [d - (+ xj] = {mH - 4a% the general formula for the^j'^ interval being a' (a- _ (_p-' xf] = {mH - 2(p- l)a'y' (5). In like manner, for the motion of the particle parallel to the axis of y, if we divide the time into intervals of which the first is equal to -5 and the rest to — ^ , we shall have for the motion n n in the (f^ interval, *' {V' - (-*" yl] = {nH -2{q-\) h'Y (6). Eliminating t between (5) and (6), we obtain as a general FREE CURVII,INEAR MOTION OF A PARTICLE. formula for the equations to the series of curves of which por- tions are successively traversed by the particle, an^ [{«' - (- ^'-' xff-v 2 Qj - 1) J = hm^ [{5' - (-«»'}' + 2(^-1)^']. As t keeps continuously increasing from zero, we must keep putting iox p and q integral values next greater respectively than ^ 'y o 2i 2 m" _mt + a 2a' 2a- ' - n~ nt + b and ,, = — ^,2 . 26' 26' Thus the formula will give us the series of equations represent- ing the successive curves. The intersections of the successive curves, if we pay attention to the signs -^'^ and -^~', wall consti- tute the limits of the portions of each which the particle actually describes. (2) To investigate the path of a particle the law of the motion of which is expressed by the integrals of the equations d'^x _ irt d/'y n^ /- n le^'^" lf^~f ^' corrected in accordance with the conditions that initially x = a, (LX (Jilt y = b, -r. = 0, -^ = ; the conjugate axes OA, OB, being sup- posed to Jdc always at right angles to Y, OX, respectively. This problem will, as we know, correspond to a path very different, after the lapse of a certain time, from the course pur- sued by the particle in the preceding problem. Integrating the equations (1), as in the first stage of the motion in the preceding problem, we have x^ = a^-~,f = f>'-'^ (2). Assume, as the most general forms of x and y, X = (cos + -' sin 6) a, y = (cos ^ + - ' sm = fXTT, 12 and, if t be greater than — , ?i~ a* . b' Suppose now that — j is less than — ; then, so long as t is less than —. . FREE CrRVILINEAR ^rOTION OF A PARTICLE. UxO U = fXTT,

m n \n m ) In the case of the equations (6), X = Q, y' = (5 cos flTT, Z = a cos jUTT, and therefore y -^--^> - =«--^' j hY~ a'z" h' a' , ^ 1 ' " -4- + -T = -4 - -i ^ ^ = 0, J In the case of the equations (7), x' = 0, y' = 0, ^' = f -^ - a' J 4 (^— - h-J . . . . (lOJ. (9> 226 FREE CURVILINEAR MOTION OF A PARTICLE. The equations (8) shew that the particle moves from its initial position to the axis of y in the arc of an hyperbola, the a* time of the motion being — , The equations (9) shew that, on m arriving at the axis of tj , the particle subsequently moves in the arc of an ellipse to the axis of z , in a time equal to -^ ^ . And the equations (10) shew that, after arriving at the axis of «', it perpetually ascends this axis, with which its path for the future entirely coincides. The position of the particle at any assigned time in each of the three discontinuous portions of its path, is given by the corresponding pair of relations between the variables and the time. ( 227 ) CHAPTER IV. COI>J STRAINED MOTION OF A PARTICLE, Sect. 1. Motion of a Particle within smooth immoveable Tuhes. Let a particle, under the action of any forces in one plane, move within an indefinitely thin curvilinear tube APB, (fig. 1 1 3J. Let X, y, be the co-ordinates of the place P of the particle, after a time t from the commencement of the motion ; and let AP = s, where A is some assigned point in the tube. Let X, Y, represent the resolved parts of the accelerating force acting on the particle parallel to the axes Ox, Oy, and aS* the resolved part along the tangent to the curve APB at P. Then the equation for the motion of the particle will be ~= X^+y'$-^S (A); clr as as or, by integration, v denoting the velocity of the particle at the point P, v' or ^, = 2f(Xdx + Ydy) + C=2fSds + C.... (B), where C is an arbitrary constant, introduced by the integration, which may be determined if we know the initial velocity and the initial position of the particle. If the force acting on the particle be a central force ; then, P representing its intensity at a distance r, we have, taking the centre of force as the origin of x, y, Xdx + Ydy = Pdr, and the formulae (A), (B), become ^'"•t («)' „' or ^ = 2jPdr+ C (D). df •' q2 228 CONSTRAINED MOTION OF A PARTICLE. (1) A particle, acted on by gravity, descends from rest down a given circular arc, the tangent to which at the lowest point is horizontal ; to compare the initial accelerating force estimated along the curve with that which would correspond to motion down the chord, when the arc is indefinitely diminished. Let A (fig. 114) be the lowest point of the arc, C the centre of the circle, P the position of the particle at any time, T the intersection of the tangent PT with the vertical line CA produced. Draw P3I horizontally, and join AP, CP. Let -F= the accelerating force at P down the arc PA, y= that down the chord PA, L CTP = a, L CAP ^ a, CA = a, AM= z, PM = y; g = the force of graAaty. Then F = g cos a, f = g cos a , -i ^-i r F cos a and tnereiore — = ^ . J cos a But, by the nature of the circle, cos a = cos Z CPM = - . a Also, observing that y^ =2ax - a^, we get , AM X ( X cos a = (^ + y) Hence I = ^ f ^> = ^^i^^^ (^^^ = (^ (2« - ^ ; f a\x J a \x J ia '\ and therefore, when x = 0, we have, for the required ratio, F=2f. Saurin ; Memoir es de VAcademie des Sciences de Paris, 1724, p. 70. Louville ; lb. p. 128. Coiu'ti^Ton, lb. 1744, p. 384. (2) The highest point of the circumference of a circle in a vertical plane is connected by means of a chord with some other point in the curve ; to determine the time in which a particle, under the action of gravity, will fall down this chord. Let AB (fig. 115) be the vertical diameter through the highest point A of the circle ; A C the chord down which the particle descends. Join BC, and let P be the position of the CONSTRAINED MOTION OF A PARTICLE. 229 particle after a time t from the commencement of its motion. Let AP = s, AB = 2a, L BA (7= a, A C= L Then, the resolved part of g, the force of gravity, along AC, being g cos a, we have, for the motion of the particle, v, = Q cos a. df "^ Integrating, we get ds — = gi cos a + C ; ds but — = 0, t = Q, simultaneously ; hence C = ; and therefore ds , -—- = at cos a. dt '' Integrating again, and observing that s = when ^ = 0, we have s = \gf cos a. Let T denote the whole time of descent down AC; then I =^\gT' cos a; but, from the geometry, I = 2a cos a ; hence 2a cos a = ^gT^ cos a, and therefore T=2('« .9, This result, being independent of the inclination of the chord to the vertical, shews that the descents down all such chords are performed in the same time ; a proposition established by Galileo. Wolff; Elementa Matheseos Tlniversce, torn. ii. p. 58. (3) From one extremity of the horizontal diameter of a circle in a vertical plane, two chords are drawn subtending angles a, 2a, at the centre ; given that the time down the latter chord is n times as great as that down the former, to find the value of a. Let j3, /3', be the inclinations of the two chords to the horizon; /, /',. their lengths, and t, i! , their times of description. Then, as in the preceding problem, it will be found that I = \g^ sin /3, I = \gt'~ sin j3', ^ ^^ r ^ t'"^ sin 3' " sin & and thereiore r = -, -: — 1\ = ^" -• — -p^ • I t sm /3 sm /3 230 CONSTRAINED MOTION OF A PARTICLE. But from the geometry it is evident that I = 2a cos /3, /' = 2a cos j3'; hence ^ = n^ -. — ^ , tan j3 = n^ tan /3' ; cos j3 sm p but it is clear that 2j3 = TT - a, 2j3' = TT - 2a, hence cot \a = n^ cot a, 2 tan \ a 1 - tan" I a 1 + cos a n tan | a, cos a and therefore cos a = ^^ , a = cos' (4) A particle is placed anywhere within a thin rectilinear tube, and is acted on by a force tending always towards a fixed centre, and varying directly as the distance ; to find the time of an oscillation. Let X be the distance of the particle, at any time t, from its position of equilibrium, r its corresponding distance from the centre of force, ju" the absolute force of attraction. Then, yJ'r being the central force at the time t, (aj jo o jo 2 5? = -"'%-=-"''' dx (Px ^ ^ clx 2 — — = - 2ux — , dt df ^ dt ' dx Let a be the mitial value of x ; then, -j- being initially equal to zero, = C-/^V; dx^ hence -^ = fi {a^ - x'). Taking the square root, we have, for the motion of the particle from its initial position to its position of rest, ^ = fidt ; (a'-xj CONSTRAINED MOTION OF A PARTICLE. 231 X integrating, we get cos"' - = C + nt; a but x= a, t = Q, simultaneously; hence C = 0, and therefore cos"' - = ni, X = a cos ut. a Now, when x acquires its greatest negative value, fit = ir ; hence, T denoting the period of a complete oscillation, we have T - - Euler; Median, tom. ii. p. 91. Cor. 4. (5) A particle is constrained to move in a straight line, and is attached to one end of an indefinitely fine elastic string, the other end of which is fixed at a distance from the straight line equal to the unstretched length of the string ; to find the time of a small oscillation. Let a = the natural length of the string, m = the mass of the particle ; 5 = its distance at any time t from its position of equi- librium, T= the tension of the string, and / = its length at the same time. Then, for the motion of the particle, d's Ts m ~r^ = — (1). df I ^ ^ Again, by Hooke's law of extension, l=a{\ +iT) (2), where £ is a constant quantity depending upon the extensibility of the string. But, s being a small quantity, I = (a^ + s'f = ail + -\, nearly ; hence, from (2), and therefore, by (1), d'^s s' / , »^ -r-r = = - - — - , nearly ; df 2eaH 2m ^ CONSTRAINED IVIOTION OF A PARTICLE. multiplying by 2 — , and integrating, we get df 4£a' ds let c be the value of 5 when — - = ; then dt = C - -^3 , and therefore dif 1/4 4x hence, supposing s to be diminishing as t increases, r. r ^h ds ,, 2a [iamy = - dt: (c^ - si put s = c COS (p, and our equation becomes 2« . i ddt j^ — Uam) = dt, c - (1 + COS' (py and therefore, 0, tt, being the values of (p corresponding to the values c, - c, of s, the time of a complete oscillation will be — (sam) Jo .. I • 2^n5 c (1 - 5 sm (p) an elliptic function of which the modulus is sin \ w. (6) A particle moves from rest from a distance a along a thin spiral tube tow^ards a centre of force in the pole attracting in- versely as the square of the distance ; to find the whole time which will elapse before the particle will arrive at the centre of force, the equation to the spiral being log - = - . r a Let IX denote the absolute force; then, by (D), we have, for the velocity at any time, ~^^-^-4 dr^C,^-^ COJ^STRATNED MOTION OF A PARTICLE. 233 but V = when r = a ; hence /I 1 V = 2fi[ y r a dr' .dS' , /I 1 df dt \r a But, from the equation to the spiral, dQ=-a'^; r hence ^^1 + „^-) = 2,. (^i - 1 j ; extracting the square root, and taking the negative sign because r decreases with the increase of t, Ave get ^ dr i J- r a ^ ^ dr i i whence (1 + aj C ^ = C - 2y^ But f dr ~ r rdr J ("i - iV i {ar - r^f ,kr(ka-r)dr^,J._Jr^^ J {ar - rj J (ar - r'f 1 1 i 2^ = - a" (ar - r^)* + i a'' vers'^ — . a Hence we have a*(l + aY (^ « vers-^ - - (ar - rj] = C- 'f,}t. Now ^ = 0, r ^ a, simultaneously; hence i7ra'(l + a^-/= C; also, r denoting the whole time of the approach to the pole, = (7-2yr; 234 CONSTRAINED MOTION OF A PARTICLE. 1 ill hence we have ^Tra* (1 + af = 2^^T, a , r y _ ira^ (1 + g-)'' 3 1 2fi (7) A particle descends by the action of gravity down a tube A (fig. 116) in the form of a semi-cubical parabola of wliich the axis Ox is vertical, and the cusp the lowest point ; to investigate the time of falling fi'om a given point A to the cusp O. Let OM=x, PM= y ; then by (B), since X = -g, Y = 0, df ^ ds Let h be the initial value of 031, then, — being initially zero, we have = - 2gh + C, ds^ and therefore -j^= 2g (Ji - x~) (1). Now the equation to the curve is ay^ = x^, I 3 i i and therefore a^y = x^, d^dy = I x''dx, ady^ = I xdx^, ads^ = \ (9a; + 4a) dx^. Hence, by (1), there is 9a: + 4a dx^ ^ ., . and therefore dt=^ ( -^ \dx, ■A \ I - X J the negative sign being taken because x decreases as t increases. Assume 2* = 9a; + 4a, and our equation becomes dt = „ . - zdz 4a z-\i 9 2(2«,)^(A.f-0 1 z^dz 3 (2agf (4a + 9h - zrf 1 z^dz , „, , , where p = 4a + 9h, 3(2agf ((B'-zi CONSTRAINED MOTION OF A PARTICLE. 235 t^ C- hh But 3 (2agJ -^ (j3' - zj r_£^ = - z (/3^- - zi +/(i3^- - ^fdz •'([i'-zj z=z- Z\ hence ^ = C - — ^-^ [-\^i^' ' ^f + ^ i3' ^i^" ^ ' ' but, initially, x = h, z' = j3^ and therefore = L_.ip^!r: 3(2«^T hence, eliminating C, we have 31' ^ = -i-Jl.o^-.=r+i/3^-cos-^l 3(2a5r)' and, substituting for z and j3 their values. When the particle arrives at the cusp, a:; = 0, and therefore the whole time of descent is equal to I 3 ah' + - (9A + 4a) cos"' ^ \ t 2 ^q7 _. . nU 3 (2a(7/ ^ (9A + 4a>'- (8) Two particles P, P', (fig. 1 1 7), of which the masses are m, m! , are connected by a straight rigid rod without weight, and being constrained to move in two straight grooves Aa, Aa, which are inclined to the horizon at given angles and are in the same vertical plane, make small oscillations ; to find the length of the isochronous pendulum. Let AP, AP', make angles a, d, with the vertical line through A, and let L PAP' be equal to «. Let T denote the 236 CONSTRAINED MOTION OF A PARTICLE. mutual action of the two particles communicated along the rod. AP = X, AP' = x, PP' = c,L PPa =(1>,L PP'a = ((>'. Then, for the motion of the two particles, we have, resolving forces along AP and AP, in —r^ = mg cos a-T cos 0, rd -j^ = mg cos d - Tcos ^'. Eliminating T, we get m cos ^' -pr - 'ni cos ^ — -j = mg cos a cos (p! - mg cos a cos ^. Cl/L etc But from the geometry we evidently have c cos = lHPT', h, fx, the absolute forces towards ^S*, H. Kesolving forces at right angles to the tangent at P, we have, by the equation (A), i2 = (^^_^^sind.+ - (1); V^-- rj p also, for the value of v at any time, there is r' r Let f, y, be the initial values of r, r , and )3 the initial value of ii; then 2^ i^ and therefore v^ = -^ -^ — - ~ — ^ + 3^ r r f f ^ Hence, from (1), we have T, /ac' ij\ . ^ 2/w,' 2//, 2/x,' 2/w. .„ V ^7 »' ^ff But 2/0 sin is equal to the chord of curvature through S, wliich, by the natui'e of the hyperbola, is equal to ; hence a P U- r'J a "" r' ^ r f f^^ _ tx {r + 2d) IX (r' - 2a) 2//,' 2/x, ^2 or' ^' 7~ 7^ = ^' (/' - 2« ) _ /^(/+ 2a) ^2 «/' ~ af ^^ «/ «/ '^^ CONSTRAINED MOTION OF A PARTICLE. 243 If the initial velocity be zero, and the particle be attracted at the commencement of its motion with equal intensity by the tAvo centres of force ; then 3 = 0, -^ = -^, , and therefore It = during the whole motion. Hence the particle would under these cir- cumstances describe the hyperbola freely. (3) A particle is moving along the convex side of an equi- angular spiral, towards the pole of which it is attracted by a force varying as any power of the distance ; to determine the reaction of the curve at any time during the motion. Let r be the distance of the particle from the pole at any time, /xr" the attractive force, a the constant angle between the curve and the radius vector, /3 the initial velocity, and a the initial value of r. Then, by the formula (A), iV being equal to /xr" sin a, we have ^2 B = ptr" sin a (l). P Again, estimating the velocity v of the particle in a direction corresponding to an increase of r, and denoting by ds an element of its path, we have y — = - /xr" cos a V-i> ds Now, by the nature of the curve, cos o ds = dr ; hence, from (2), dv dr integrating and observing that /3, a, are the initial values of v, r, we get 2ll ^ ^~ _^S' = -^ (r-^ - a-0 (3)- « + 1 Again, p denoting the perpendicular from the pole upon the tan- gent to the curve, we have, since p = r sin a, dr r . . P = r — - -. (4). ' dji sin a From (1), (3), (4), we obtain . sin a (p., 2(j. . ,1 ^,,1 1 r [_ n + I J n +3 „ . /3' sin a 1y. sin a ?^+ 1 r (w + l)r Euler; Median, torn. ii. p. 86. r2 2U CONSTRAINEJ) MOTION OF A PARTICLE. (4) A particle, starting from rest, descends down the convex side of a circle from a given point in its circiuiiference ; to find where it will leave the curve. Let (fig. 121) be the centre of the circle, AO being a ver- tical radius. Let P be the initial position of the particle, Q its point of departure ; P3I, QN, horizontal lines. Join OQ, and let Z AOQ = cp ; a = the radius of the cucle. Then, the centrifugal force at Q being equal to the normal component of gravity, wc have - = y cos (^ ; but, denoting 3IN by x, c^ = 2gx ; hence, putting 310 = c, 2x = a cos (p = c - X, X = ^c. Fontana; Memorie della Societa Italiana, 1782, p. 175. (5) A particle acted on by gravity oscillates in a chcular arc ; to find the reaction of the curve at any point. Let (fig. 122) be the centre of the circle ; P the position of the particle at any tune ; A the lowest point of the circle ; L A OP = Q. Then, B being initially equal to a and the velocity zero, we have R - g{Z cos Q - 2 cos a). (6) A particle descends down the convex side of a logarithmic curve placed with its asymptote parallel to the horizon ; to find where it leaves the curve. Let P (fig. 123) be the point in which the particle is placed, and Q the point of its departui-e ; 0J/= A, 0N= x. Then, the equation to the curve being y = log„ x, we have, putting A = log^ a, x = jhh + ii + A%'y\. Fontana; lb. p. 182. (7) A particle descends from rest down the convex side of an ellipse with its major axis vertical, from a given point in the curve ; to determine where it will leave the ellipse. Let the highest point of the ellipse be taken as the origin of co-ordinates, the axis of x being vertical, and that of y horizontal. CONSTRAINED MOTION OF A PARTICLE. 245 Let a, b, denote the semi-axes major and minor ; h the initial dis- tance of the particle from the axis of y, and x the distance of the point at which it leaves the curve. Then the value of x will be a root of the cubic equation {a" - V){x^ - 3a.r) - ^a^^x + a' {b- + 2ali) = 0. Fontana; lb. p. 175. (8) A particle descends from rest down the convex side of the Cissoid of Diodes, which is so placed as to have its asymptote vertical ; the initial place of the particle being known, to find the point in which it will leave the curve. Let P (fig. 124) be the initial position of the particle, and Q its place on leaving the curve. PS, QN, at right angles to Ox, Oy; PS=h, QN = x. Then, a bemg the radius of the gene- rating circle, the value of x will be a root of the cubic equation , 16a , 64a- + 3 6 A' 8a/r ^ X X' + X • = 0. 9 81 9 If the motion commence at the cusp 0, h = 0, and therefore 8 X = ~ a. ^ Fontana; lb. p. 181. (9) A particle is projected with a given velocity along the convex side of a parabola from a given point in the curve ; to determine the reaction of the curve at any time of the motion, the particle being always attracted to the focus by a force varying inversely as the square of the distance. Let S (fig. 125) be the focus of the parabola; B the point from which the particle is initially projected \n the direction of the tangent BT ; P the position of the particle after any time, SP = r, SB = a, SA = m, j3 = the velocity of projection, /x = the absolute force towards S. Then, at P \ry \a 2 (10) A particle is projected with a given velocity at the highest point of a circle in a vertical plane along the concave side of the curve ; to determine the pressure on the curve at any point in its path. 246 CONSTRAINED MOTION OF A PARTICLE. Let AOB (fig. 126) be the vertical diameter, being the cSiitre of the circle ; P the position of the particle at any time, OP = a, lAOP = 0; j3 the velocity of projection at A ; then, for the pressure at P, i2 = '— + ^ (2 - 3 cos 9). Suppose that jR = initially ; then — = g, and H = 3(/ vers = Qg, when = tt ; which shews that when the particle arrives at the lowest jjoint the reaction is six times the force of gravity. Euler; Median, torn. it. p. 65, Cor. 7. (11) A particle moves along the convex side of an ellipse under the action of two forces tending to the foci and varying inversely as the square of the distance, and a third force tending to the centre and varying as the distance ; to find the reaction of the curve at any point. Let P, denote the reaction of the curve on the particle at any point, jo the radius of curvature ; f,f' the initial focal distances, and />(,, yl the corresponding absolute forces ; //, the absolute force to the centre, 2a the axis major of the ellipse, and /3 the initial velocity. Then If j3', /3', j3' , denote the velocities which the particle ought to have initially to revolve freely round the three centres of force taken separately, and therefore, when the forces are taken conjointly, it "will revolve about them freely when (12) An ellipse is placed Avith its major axis in a vertical position ; to find the A'elocity with which a particle must be pro- jected vertically upwards from the extremity of the minor axis CONSTKATJS'ED MOTION OF A PARTICLE. 247 along the interior of the elliptic arc, so that after quitting the curve it may pass tlirough the centre. If a, h, denote the semi-axes major and minor, the required velocity will be equal to Sect. 3. Inverse Problems on the Motion of a Particle along immodeahle plane Curves. (1) To find a curve EPF (fig. 127) such that, A and B being two given points in the same horizontal line, the sum of the times in which a particle will descend by the action of gravity down the straight lines AP, BP, may be the same what- ever point in the curve P may be. Bisect AB in O ; let Ox, a vertical line, be the axis of x, and OAy, which is horizontal, the axis of y ; let AB = la. Then, X, y, being the co-ordinates of P, the times down AP, BP, will be respectively equal to { x~ + (« - yj \\ f x' + (a + yf Hence, k denoting the sum of the times, (1 J^gxf = {x' + (a- yyy + {x' + (a + yJY (1). Putting 2 k~y = Ac, and squaring both sides of the equation, we have 2cx = a' + X' + y" -i- [x- + (a - yJY [x' + (« + y)'}*, {2cx - a' - X' - y'f = (a;^ ^- {a - yf] {x' + (a + yj}. Developing both sides of the equation, and simplifying, we shall readily find that c'x^ - a~cx - cx^ - cxy' = - a^y'^, and, therefore, x^ - ex + a^ y' = ex —^ (2), '' a - ex which is the equation to the required curve. If we trace this curve, we shall find it to consist of a branch VOV having an asymptote parallel to the axis of y, and of an 248 CONSTRAINED MOTION OF A PARTICLE. oval EFP. The oval is the portion of the curve which corre- sponds to the problem which we are considering. The infinite branch VO V would correspond to the condition that the times down -AP" , BP" , shall have a constant difference : in which case we should have had, instead of the equation (1), (5 ^""(j^y = [x' + (« + yJY - {^' - (a - yfY; whence, by the involution, we should have obtained the same equation (2). The curve has pretty much the shape of the Conchoid, although its equation is ess.entially different. Fuss; Memoir es de V Acad, de St. Petersh. 1819. (2) A particle, not acted on by any forces, is constrained to move within a thin tube of such a form that the acceleration of the particle parallel to a given straight line is invariable ; to determine the equation to the path of the particle. Let the axis of x be taken parallel to the given line ; let c be the constant acceleration of the particle parallel to the axis of z, and )3 its velocity within the tube, which wiU be invariable. Then .jx' ,iy>. d'^x but, by the condition of the problem, -— = c, and therefore, the dx axis of y being so chosen that j^- = when a: = 0, „ dx d^x „ dx da? , , Eliminating dt between the equations (1) and (2), we get dy f'2a - x\ or, putting — = a. dx \ X whence, by integration, the position of the axis of x being sup- posed such that X = when y = 0, , - X y = (2ax - x'f + a vers"^ - ; a CONSTRAINED MOTION OF A PARTICLE. 249 which is the equation to a cycloid of v.'hich the axis is parallel to the given line. There is an elaborate investigation by Euler, in the Meynoires de V Academie de St. Petersb., torn x. p. 7, on the nature of the curve of constraint when the particle is subject to the action of gravity, and the direction of uniform acceleration is horizontal. A notice of this problem may be seen in the Bulletin des Sciences de Bruxelles, torn. ix. (3) To determine the curve down which a particle may descend by the action of gravity, so as to describe equal vertical spaces in equal times, the tangent to the curve at the point where the motion commences being vertical. Let (fig. 128) be the j)oint where the motion commences, Ox the axis of x touching the required curve OA at 0, Oy the axis of y at right angles to Ox ; 031 = x, P3I = y, j5 = the invariable velocity of the particle parallel to Ox. C+ 2gx. (7+ 2gx But when x = Q, -—- = ; and therefore dx dx~ ^ ' y^ 2.^' ' no constant being added because x = 0, y = Q, simultaneously. The required curve OA is therefore the semi-cubical parabola, O being the cusp, and Ox the axis. This curve is called the Isochrone. It was proposed by Leib- nitz,* as a challenge to the disciples of Des Cartes, who, from an • Nouvelks de la Republiquc des Leitres, Septetnbre 1687. Then ds' df = c + 2^^, C beiu! ds' dx"^ dx' df But dx It " /3; hence &' = 4/3^ ,djl_ dx' 250 CONSTRAINED MOTION OF A PARTICLE. excessive attachment to the geometry of their master, aiFected to despise the methods of the Differential Calculus. No solution was communicated by any of the Cartesians. Huyghens alone successfully accepted the challenge, by whom a geometrical solution was given in the Nouvellcs cle la Rejnihlique des Lettres, Octohre 1687. The solution by Leibnitz appeared for the first tixae in \he Acta Erudit., Lips. 1689,p. 196,etsq. The solutions both of Huyghens and of Leibnitz were synthetical. An ana- lytical solution was given afterwards for the first time by James Bernoulli.* (4) A particle is projected with a given velocity from a point A (fig. 129) along a horizontal line AO towards a point ; to find the curve along which it must be constrained to move that it may approach the point O uniformly ; the particle being acted on by gravity, and A being a tangent to the required curve. Let P be the position of the particle at any time ; AO = a, OP ^r, lA op = 9 ; (3= the velocity of the particle at its initial position A. For the motion of the particle at any point in its descent there is dr' odO' r>^ ^ a df-''"df = ^'-'^^'''''^' dr^ /, o dO-\ ,1, ^ . fl df\ drj ' '' But, by the condition of the problem, — = (7, a constant HZ quantity : hence JO But, initially, 6=0, r — = ; hence C^ = /3^ and therefore 7/12 • I5S'~ —-i = 2^^^ sin 6, dr^_^ ^ . 7'" {2gj (sin ej * Jet Erudil., Lips. 1690, p. 217. COXSTKAINED MOTIOX OF A PARTICLE. 251 integrating, and observing that 0=0, r = a, initially ; we have 2 (2y)' J, (sine/ which is an equation for the construction of the path of the par- ticle. The particle will move from A along ABCO to the point Avith a uniform velocity of approach ; it will afterwards move from along OcBa with a uniform velocity of recession. When it has arrived at a it will proceed uniformly along Oa produced. This curve has been called the Paracentric Isoclu'one by Leibnitz, by whom the problem was originally proposed as a challenge to the mathematicians of the day, in the Acta ErudiL, Lips. 1689, p. 198. Several years elapsed before the problem received a solution. At length James Bernoulli succeeded in obtaining one, which appeared in the Acta Erudit., Lij)S. 1694, p. 277. Solutions were shortly afterwards published by Leib- nitz and John Bernoulli, in the Acta Erudit., Lijis. 1694, p. 371, 394. The problem was afterwards generahzed by Varignon in the Memoires de V Academic des Sciences de Paris, 1699, p. 9, et sq. (5) To find the nature of the curve OP A (fig. 130) such that a particle acted on by gravity' will descend down any arc OP in the same time as down its chord. Let Ox be vertical, Oy horizontal, PM parallel to y 0. Let 0P= r, LxOP= 9, arc OP = s, 031= r cos 0. Then, since the velocity acquired down the arc OP is the same as that which is due to falling freely down 031, ds' n 7 ^ ^^ — ., = 2yr cos tf, dt = ^ ; (2yf (r cos of and therefore the whole time of descent down OP is equal to 1 r ds (2yfJ, (r cos &f But the time of descent down the chord OP is equal to yj \cos 9 252 CONSTRAINED MOTION OF A PARTICLE. and, therefore, by hypothesis, '2\i/ r \h 1 C' ds ^ ^ ^ (2^JVo C^- COS Of 2- ■ '' ■ ^" Vcos e) I (r cos oy DifFerentiating both sides of the equation, we have ''cos 0\ cos d dr + r sin Bd9 ds (cos Of , « ' ' ^ (r cos &) X COS <7r + r sin t/0 = cos {dr + r'dB^f. Squaring both sides and simplifying, 2 sin B cos rc?r(/0 = r^ cos 20 c/0', c/r cos 20 -„ — = -. dtj. r sm 20 Integrating, log r^ = log a^ + log sin 20, r"^ = a^ sin 20, where d^ is some constant quantity. From O draw OE, bisecting the angle xOy, and let L POE^cp; then, since = 4 tt - 0, we have r = a" cos 20 ; which is the equation to the Lemniscata of James Bernoulli, O being the centre and A the vertex of the equilateral hyper- bola. This very beautiful problem is due to Saladini. Saladini ; Mctnorie dclV Istituto Nazionale Italiano, tom. I. parte 2. Fuss ; Memoires de PAcad. de St Petersb. 1819. (6) To find the equation to the tautochrone when a particle is acted on by any forces whatever in one plane. A tautochrone is a curve along which a particle acted on by any assigned forces will arrive in the same time at a given point from whatever point in the curve its motion commences. Let A (fig. 131) be any assigned point, and E any point whatever in the curve AEB ; then the time from E \.o A\% to be indepen- dent of the position of E. CONSTRAINED MOTION OF A PARTICLE. ^53 Let P be any point in AE ; AP = s, AE = a, S= the sum of the resolved parts of the accelerating forces on the particle along the tangent P J* at the point P. Then, for the motion of the particle, ^.^ ,7,2 But, the particle being supposed to have no initial velocity, = C - 2j''Sds ; and therefore ^r^ = 2 Sds (1), The time from jB to ^ is equal to 1 C ds Sds f and this formula must be independent of a. Hence we must have r ds /s' Sds''' '" where ( - ) denotes some function of - . Hence ^'u)' f^''=j7m''' Sds , , ^y y-^jj and therefore, differentiating with respect to s, Sds = d ' s' .«> But the tautochrone AB being an invariable curve, whatever be the value of a, it is manifest that « must not appear in this equation ; hence J 0' I - J I =: — — , where ^ is a constant quantity, 254 CONSTRAINED MOTION OF A PARTICLE. and therefore S= /cs (2), k being some constant quantity. Hence, by (1) and (2), ;^ = /.(<■-'';. and therefore, if t denote the time the motion from E to A, 1 C" (Is k Hence we have, from (2), S = \ TTt'S, which is a differential equation to the tautochrone. The direct problem of Tautochronism in the case when gravity is the accelerating force, was fii-st considered by Hiiy- ghens, in his Horolog. OscilL, where he proves the inverted cycloid with its axis vertical to be tautochronous. The inverse problem was first considered by Xewton, P;v";?c?/?. Hb. i. sect. 10. See also Eider, Comment. Petrop. 1729, and Mechan. torn. ii. p. 211. (7) A particle is acted on by an attractive force tending towards a fixed centre, and varying as the distance ; to find the tautochi'one. Let /w, denote the absolute force of attraction, r the radius vector at any pomt of the cm-ve, 2^ the perpendicidar from the pole upon the tangent at the point, the inclination of the tangent to the radius vector. Then, by the formida of the preceding general problem, we have, putting />tr cos for *S', 1x7' cos = 5 TT'tS, whence (xd (r cos <}>) = ] Tr'r'ds ; but ds cos (p = dr ; hence we have fxr cos (p d(r cos (f) = \ Tr'rrdr ; integrating, we get ij.r cos- + (7= ]7^W^ or tx (r -p') + C =\ TT^rr'. CONSTRAINED MOTION OF A PARTICLE. 255 Let c be the value of r when s = 0, and therefore when = 2 TT ; then also p = c, and consequently C=l7rW. Hence pt (r - y) + 5 tt^c't = 5 tt W", which is the differential equation to the curve. Euler; Median, torn. 11. p. 208. (8) An infinite number of similar curves originate at a given point ; to determine the corresponding synchronous curve, or the curve which shall cut them in such a manner that a particle acted on by gravity may describe the intercepted arcs in equal times. Let (fig. 132) be the given point, and CPD the synchro- nous ciu've intercepting the arc OP of the curve OPQ, which is one of the similar curves. Let Ox, a vertical line, be taken as the axis of x, and Oy, at right angles to it, as the axis of y. Let 031= X, PM= y, OP = s. Then, if k denote the time down OP, which by hypothesis is constant for every point P in the curve CPD, we have j-j^^.ni±4,^ a, where p is equal to -^ . ax Now, by the nature of similar curves, the equation to the curve OPQ is of the form jg,2)-o, o.,.«/g) (2), where F, f, denote certain functions of the quantities to which they are prefixed, a being the value of the general parameter of the class of similar curves for the individual cui've OPQ. Hence, assuming X = cir, and therefore y = of{r), by (2), .... (3), we have, from (l), 1 rT(\ , T-f '- k^a' \ ^-"^±^^1 dr = d'{r) (4), 266 CONSTRAINED MOTION OF A PAKTICLE. where T= -^ = — /(r), and (t) is some function of t. Hence, from (3) and (4), there is Eliminating t between these two last equations, we shall obtain an equation in x, y, the required equation to the sychronous curve. If the integration indicated in the equation (1) can be effected, then it is needless to have recourse to the subsidiary sjaiibol r. We have merely in this case to eliminate, after the performance of the integration, the parameter a, by the aid of the equation (2). It rarely happens, however, that we can execute the operation of integration, and under these circumstances the equations (5) will enable us to construct the synchronous curve by the method of quadratures ; a pair of values of x, y, and therefore a point in the synclu'onous curve, being ascertained approximately for every numerical value which we may assign to T. The problem of Synchronous Curves was first discussed by John Bernoulli, in the Act. Eruclit. Lips. 1697, Mai. p. 206. The subject was afterwards investigated by Saurin, and by Euler.^ (9) An assemblage of circles in the plane xOy, (fig. 132), all touch Ox in the point O ; to determine the sychronous curve. Ox being vertical, and gravity the accelerating force ; the descent being supposed to commence from O. The equation to any one of the circles, its radius being a, will be x^ = lay - y, or ?/ = « [l-f l---„Vl r Adopting the notation of the preceding general problem, we ^^^'^ f{r)=\-{l-ri, ' Median, torn. ii. p. 47; Mem. de VJcad. de St. Pelersb. 1819— 1S20, p. 20, 35. CONSTRAINED MOTION OF A PARTICLE. 5;,0 ( dr Hence .^^0^/—,, , . ^^^' (' -(1 - f } , whence the requii'ed cui've may be constructed by the method of quadratures. Euler; Median, tom. ii. p. 52. (10) A particle acted on by any assigned accelerating forces in one plane moves along a curve from one given point to another ; to determine the form of the curve, that the whole time of the motion between the two points may be the least possible. Let P (fig. 133) be any point in the required curve ; 0M= x, PM= y ; A and B the two given points, AP =■ s ; also let a, (5, be the values of x at the points A, B. Then, v being the velo- city of the particle at P, 1 dt = — = -^ ^-^ dx, where « = -^ , V V dx and the w^hole time from A to B will be equal to f(li^%. (1). i Assume ^-^ = V; then, that the expression (l)may be a minimum, we have, by the Calculus of Variations, since F in- volves only p and v, of which the latter is a function of only x and y, ,7P N-'If^O (2), dx where iV, P, denote respectively the partial differential coeffici- dP ents of V with regard to y,p; -y- representing the total differ- ential coefficient of P with respect to x. Now ^^ 1 ,, J dv ,„, 258 CONSTRAINED MOTION OF A PARTICLE. where -r- signifies the partial differential coefficient of v with dy regard to tj ; but ^^„ ^ xdx + Ydy (3), where X, Y, represent the resolved parts of the whole accele- rating force on the particle parallel to Ox, Oy ; and therefore, in (3), — = — . Hence dy V N=- Y^(i+pi = -^^. v^ V dx Again, P^^^^l%. Hence, substituting for N and P in (2), we have F ^ ^ _ (3). From (1), (2), (3), ■ P = - we have \a 1 dp ' ' A J _ fx p dr ''Tp / 2 \ '1 r \ p dr ) r dp' 2± = - p r adr (r - a) ' integrat ing. we 1 get 2lc )gp = log e+iog- - a 3 r logp'^logfc'-^ but C must be a negative quantity, because, as wiU appear from the equation (2), a is greater than r ; hence, putting - A for C, CONSTRAINED MOTION OF A PARTICLE. 261 we have, for the differential equation to the brachystochronej r If from this equation we were to obtain, by integration, a relation between r and an angular co-ordinate 0, we should introduce another constant into the equation in addition to A. Both these constants would have to be determined by the con- ditions that the curve must pass through both A and B. Euler ; Median, torn. ii. p. 191. (12) To find the inclination of a thin tube to the horizon, so that a descending particle may describe the greatest horizontal space in a given time. Required angle of inclination = 45°. (13) A particle having been placed at the point A, (fig. 135), moves along a thin tube APS towards a centre of attractive force in *Si which varies as any function of the distance ; to find the nature of the curve of the tube that the time through any arc AP may be n times as great as through a portion Ap of the prime radius vector SA, Sp being equal to SP. Let *SP = r, SA = a, L ASP = 6 ; then the equation to the curve will be (14) A particle is projected with a given velocity from a point A (fig. 136) along a curve APO in which it is constrained to move, and is acted upon by a force always tending to 0, and varying dhectly as the distance ; to find the nature of this curve in order that the angular velocity of the radius vector OP may be invariable. Let AO = a, OP = r, lAOP = B, /J = the absolute force of • attraction, w = the angular velocity of OP, j3 = the initial velocity of the particle ; then the equation to the curve will be Euler ; Mechan. tom. 11. p. 138. (15) A particle is acted on by an attractive force tending to a ( 262 CONSTRAINED MOTION OF A PARTICLE. centre, and varying inversely as the square of the distance ; to find the tautochi-one. If T denote the time of the motion, and the notation remain the same as in problem (7), the differential equation to the tautoclirone will be -.2 2 /> =r - -—.{r -c)r. 2juc Euler; Median, tom. 11. p. 209. (16) To find the tautochrone when the central attractive force is constant. If f denote the constant central force, the equation to the tau- tochrone will be „ /, Tf'TC\ , -n^r , Euler; Mechan. tom. 11. p. 210. (17) An infinite number of straight lines originate at a single point and lie in one plane ; to determine the synchronous curve, gravity being the accelerating force. The given point being taken as the origin of co-ordinates, the axis of X extending vertically downwards, and that of y being horizontal ; the synchronous curve will be a ciixle of which the equation is ^ + y' = ^ gJi'x, where k denotes the common time of descent. Euler; Mem. de I'Acad. de St. Petersh. 1819, 1820, p. 22. (18) There is an infinite number of cycloids, of which the bases all commence at the origin of co-ordinates, and coincide with the axis of y, which is horizontal ; to find the synchi-onous curve, gra-vdty being the accelerating force, and the motion com- mencing from the origin. Let k denote the constant time of descent ; then, the axis of X being vertical, the equation to the required curve depends upon the elimination of a between the two equations r f(i\^ X - X =■ a vers"^ {^ [-]} , V = « vers"^ — (2ax - x^f, I \aj J a and will cut all the cycloids at right angles. John Bernoulli; Act. Erudit., Lips. 1697, Mai. p. 206. (19) A particle, acted on by a central force attracting directly as the distance, moves along a curve from one given point to CONSTRAINED MOTION OF A PARTICLE. 26S another; to find the nature of the curve when it is brachys- tochronous. Let A (fig. 134) be the point where the motion commences, and B the point where the particle is to arrive in the shortest time possible. Let P be any point in the brachystochrone ; SP = r, p = the perpendicular from S, the centre of force, upon the tangent at P, a = SA. Then the equation to the curve be- tween jw and r will be p^ = A (r^ - a^) where ^ is a constant quantity, which is the equation to the hypocycloid. If from this equation we were to obtain by integration a rela- tion between /• and an angular co-ordinate 6, we should have another constant in the equation in addition to A. Both these constants would have to be determined by the conditions that the cui've must pass through both A and B. Euler; Mechan. torn. ii. p. 191. Sect. 4. Inverse Problems on the Pressure of a Particle on Smooth Fixed Curves. (1) A particle descends down a curve line in a vertical plane by the action of gravity; to find the nature of the cui've that the pressure may be invariable. Let OA (fig 137) be the required curve; Ox, vertical, the axis of X, Oy, horizontal, the axis oi y; P any point in the curve, OM = x, PM = y, OP = s ; k the constant pressure ; /3 the initial velocity of the particle, being its initial position. Then, by formula (A) of sect. (IL), we have , dy \ ds^ /,. ^'^ds^-.lf «' where p denotes the magnitude of the radius of curvature at P. ds^ But —=(i- + 2gx; also, s being taken as the independent variable, d'^y ! = !£ p dx ds 264 CONSTRAINED MOTION OF A PARTICLE. Hence, from (1), we have ds k dz , ^„ 1 d^y a dxi dx " -5- = {2gx + &) —4 + ^ i- -7- (2gx + ^-f (2gx + ^'f Integrating, we have dy k C ^(2gx-,(3i=^(2gx + (3i^^ + C, where C is an arbitrary constant. Assume a to be the inclination of the curve to the vertical at the origin ; then and therefore. k C sin a = - - 3 ; ^ = ^ _ 2 ^~^ ^^" " . . . (2). '^* ^ ^ {2gx + /3'f The relation between x and y may be obtained by a second integration, but the result is of little value in consequence of its complexity. For the investigation of the form of the curve which corresponds to the differential equation (2), the reader is referred to Whewell's Dynamics, part 11. p. 95 ; or, Earnshaw's Dynamics, p. 129. The problem of the Curve of Equal Pressure, in the case of gravity, was first proposed by John Bernoulli^ and solved by L'HopitaP. Various problems of a similar character were after- wards discussed by Varignon'. Commerc. Epistolic. Leibnitii et Bernoullii, Epist. vii. (2) A particle, acted on by gravity, descends from a point O (fig. 137) down a curve OA, which it presses at each point of ' Act. Erudil. Suppl. torn. ii. Sect. G. p. 291. * Mem. de I' Acad, des Sciences de Paris, 1700, p. 9. ^ Mem. de VAcad. des Sciences de Paris, 1710, p. 196. CONSTRAINED MOTION OF A PARTICLE. 265 its descent with, a force varying as the square of its distance below the horizontal line through ; to find the nature of the curve OA, the initial velocity of the particle being zero. Let the axes Ox, Oy, be taken vertical and horizontal ; let k be the pressure on the curve when x is equal to unity. Then, by the formula (A) of Sect. (II.), •^ ds p but hence ^ dx d^x dy dy^ _ integrating, we have no constant being added because the curve passes through the Putting y = «, we get k 1 (25a^ - x*f dy = x'dx, which is the equation to the Elastic Curve of James Bernoulli^ Varignon ; Memoires de VAcademie des Sciences de Paris, 1710, p. 151. (3) A particle, acted upon by a force parallel to the axis of x, is constrained to move along a given curve OP A (fig. 137); to find the law of the force that the curve may experience an inva- riable pressure. 1 Act. ErtuUt., Lips. Wn, p. 272 ; 1695, p. 5-38. 266 CONSTRAINED MOTION OF A PARTICLE. Let k denote the constant pressure, /3 the velocity of the particle at 0, which we ■nill take as the origin of co-ordinates, and X the force, at any point P of the curve, parallel to Ox. Then, by formula (A) of section (II.) and formula (D) of section (I.), we have A- = X ^ + -^ {j3^ + 2 rXdx]. as p ^ Jo Taking s as the independent variable, we have d'y p dx ds and the equation becomes A; ^ = 6^ ^ + X - ^ + 2 ^ r Xr/:r. ds ds' ds ds c7sVo Multiplying by -j- ds, and integrating k {'h 'II ds==h & ^' + ^ r X dx, j ds ds ' ds^ ds^ Jo ds' and therefore, putting -f- = p, rxdx=-^,(5Ukf'.A{j^^.c (1> Differentiating with respect to x, we obtain the required ex- pression for the force If we put x= 0, we have, from (1), 2kl\+ 1 '■'^(1 +py ^ CONSTRAINED MOTION OF A PARTICLE. 267 a condition which will determine the value of the arbitrary constant C. Euler ; Median, torn. ii. p. 101. (4) A i:)article moves along a parabola OA, of which the axis is Oy, under the action of a force always parallel to Ox which is at right angles to Oy ; to determine the law of the force that the particle may exert the same pressure on the curve during the whole of its motion. Let Ox, Oy, be the co-ordinate axes, k the constant pressure, and x^ = ay the equation to the parabola. Then, by the formula 2x for X given in the preceding problem, since ^j = — , we have where C is a constant quantity, = -^C-—fa'- 2x')(a' + ^xj . . . .(1). Again, by the formula (1) in the preceding problem, Xdx = - Ii3' +^fe + l) IK^' + 4:^7 + C], Ax' rX dx = - 2j3V + 7a) £-'"' = (wV- + /3- - w-«'/ - a>r (2). Hence, by (1) and (2), we have 2a>r = (i3 + wa) e'"* - (i3 - a;«) s"'"', which gives the value of r at any time during the motion. The equation to the path of the particle is, putting 6 = wt, 2oji- = (/3 + wa) £^ - O - cjo) H-". John Bernoulli; OjJet^a, torn. iv. p. 248. Clairaut; Mem. Acad. Paris, 1742, p. 10. (3) A material particle is placed within a thin circular tube which is constrained to revolve with a uniform angular velocity in a horizontal plane about a point in its circumference ; to investigate the motion of the particle. Let (fig. 142) be the point about which the circle APO is constrained to revolve ; C its centre at any time t, and P the position of the particle ; R the action of the circle on the particle, which will take place in the direction PC. Let Ox, Oij, be the axes of co-ordinates, x, y, being the co-ordinates of P. L POx =e, A OPC= L COP = <}>, 0P = r, 0C= a. T 274 CONSTRAINED ^[OTION OF A PARTICLE. Then, since no accelerating forces act on the particle, we have, by the formulae (A), J= - i? cos(0- ^), ^--H sin (0-0) (1); multiplying these equations by sin (0 - (p), cos (0 - 0), and sub- tracting, we have or, since x = r cos 9, y = r sin 9, sin {9 - (p) —2 (r cos 6) - cos i9-) = 0, sin(0-0)— ,{cos(0+0)+cos(0-0)}-cos(0-0)-^{sin(0+0>sin(0-0)}=O But, supposing w to be the angular velocity of the diameter OCA of the circle about 0, and L AOx to be initially zero, it is clear that /- AOx or 9 + (p = wf (3). Hence, putting wt for 9 -^ (p, sin (9 -(f) -2, cos {B-) -p sin (0 - 0) + sin (9 - d)) -t:; cos wt - cos (0 - d») -y^ sin w^ = 0, ^^ df dtr ^jsin (0 - 0) ^^ cos (0 - .^) - cos {B - i>) 1^ sin (9 - A - w" sin (9 - 0) cos o)t + 10- cos (0 - 0) sin w^ = 0, _ ^^ (0 _ 0) _ o^'sinCe _ - wO = 0- But, by (3), we have B = wt - (f) ; hence 2 ^ + w^ sin 2d. = 0. or CONSTRAINED MOTION OF A PARTICLE. 2T5 Multiplying by 2 -^ , and integrating, 2 'i|J - c^^ cos 2rf, = O (4). For the sake of simplicity we will suppose that initially P coincides with A, and that its velocity is zero ; hence when ^ = we have = 0, and since from (3) eld d(h 1 — ~ = (i), dt dt we have also initially —^ = w. Hence, from (4), 2w^ - w^ = C, and therefore 2 -^ - cos , O/ tk ^ (m + m ) -rn; - ma -^^ cos d = 0, ^ ^ df df d^x , d^ . m sin B -TT - nia cos -7^ sin = : dt df eliminating -r-j between the two last equations, 72 72 (771 + m!) cos -^-^ sin B - m sin d ^r-- cos B = Q ; dt dt' multiplying by 2 — , and integratuig, (m + rd) [ — sin J + m ( — cos ) =C, dt dB~ {m + in cos- B) j-^ = C; dB but, initially, B = a, — = w ; hence do (ni + m' cos" a) w" = C, - , „ c/0^ o ni + m cos'^ a .. and thereiore -^v = w' ; —^ {8). dr m + m cos B Again, integrating (4), we get and therefore, by (6), 7)1 —- + m —-= C, df dt m + 711) -rr + 771 a svn B ^r ^ ^ j ^ c?^ dt but, at the commencement of the motion, -^ = B, B = a, -^- = w; dt dt hence (ni + m') /3 + m'aw sin a = C, , ,T ^ dx r> TTilau) sin a w'a sin B dB and thereiore -7- = p + ; — -j- ; dt m-V 771 771 + 771 dt whence, by (8), dx ^ 7n'au) sin a 7n'ao} sin B (Trn + 7n co^ a\l /q\ ;7- = /3 + -, — ; 271 ^ ■>' dt 771 + 771 771 + 771 \77l + 771 COS BJ 278 CONSTRAINED MOTION OF A PARTICLE. The equations (8) and (9) give us the velocity of P along Ox, and the angular velocity of Q about P, for any assignable inclina- tion of the rod to the line Oz. If between these two equations we elmiinate clt we shall obtain a differential equation to the path of Q in x and 0. From (8) we have w ff . ^\ r. t = I (w^ + m' cos" OydO, (m + m cos" a)* an elliptic transcendent for the determination of t for any value oiB. Clairaut; Mem. de VAcad. des Sciences de Paris, 1736. p. 10. (5) A string is completely coiled round the circumference of a circular lamina, and has a particle attached to one exti'emity which is free, the other extremity being fixed to the lamina : every particle of the lamina repels the free particle with a force varying inversely as the distance ; to find the velocity of the particle at any time after its departui"e from the circumference of the lamina. Let a denote the radius of the lamina, r the distance of the particle from its centre at any time, and f the initial repidsive force experienced by the particle. Then, as may be ascertained by the performance of the appropriate integrations, the repulsive force on the particle at any time from the centre of the lamina \^'ill be — . Hence the particle may be considered as moving along a curve which is the locus of the free extremity of the string acted on by a central repulsive force -^ ; and therefore, by the formula (D) of section (I), t-=C+2{^I dr = C + a/log r' = C + af log {p- + a"), if p = the length of the string set free. But, initially, r = 0, /o = ; hence Q = C + af log a'. CONSTRAINED MOTION OF A PARTICLE. 279 and therefore 1? - af log ~ — 5 — . Let Q denote the angle subtending the arc of the circum- ference of the lamina from which the string has been unfolded ; then p = aO, and we have v' = aflog{l + e^ (6) Two particles connected together by a rigid rod without weight are projected along a smooth horizontal plane ; to deter- mine then- motion. Let the plane of co-ordinates coincide with the plane of the motion. Let ?n, n, be the resolved parts of the initial velocity of the centre of gravity of the two particles parallel to the axes of z, y, and let a, b, be its initial co-ordinates. Let w be the initial angular velocity of the rod, B its inclination to the axis of x at the end of the time t, and £ at the beginning of the motion. Then the position of the centre of gravity is given at any time t by the equations X - mt + a, y = 7it + b ; and the inclination of the rod to the axis of x, by the equation V = (i)t + Z. Clairaut ; Memoires de I' Academic des Sciences de Paris, 1736, p. 7. Euler; Act. Acad. Petrop. 1780, P. 1. ; Opuscula,De motu corporum fiexihilium, torn. iii. p. 91. (7) A spherical particle moves within a smooth tube which revolves about one extremity with a uniform angular velocity in a vertical plane, the capacity of the tube being just sufficiently great for the reception of the particle ; to determine the motion of the particle. Let Ox, (fig. 140), which is horizontal, be the initial position of the tube, and P the position of the particle in the tube after a time t. Let w denote the angular velocity of the tube about O, 6 the inclination of OP to Ox, and OP = r. Then, supposing the initial velocity of the particle to be zero, and that r = a initially, the value of r at any time t is given by the equation r = -^ sm u)t + -^ c"" + ^ f "" , 2w' W 4w- ~80 CONSTRAINED MOTION OF A PARTICLE. and the polar equation to the path of the particle will result from the substitution of 6 for o)t in tliis equation. "When t becomes very great, the jDolar equation becomes r = ^ e^ which is the equation to an equiangular spiral. The solution of this problem was attempted by M. Le Barbier, in the Annales de Gergonne, torn. xix. p. 285, who omitted to take into consideration the centrifugal force, an oversight which entirely -^dtiated his results. The correct solution was given in tom. XX. by Ampere. (8) A material particle P (fig. 144) is fixed to one end of a rigid rod PQ without weight lying upon a smooth horizontal plane. The end Q is constrained to move with a uniform velo- city in the cii'cumference of a circle ABQ; to find the velocity of the increase of the angle PQR, O being the centre of the circle, and OQR a straight line. If PQ = h, OQ = a, L PQR = i// at any time t, a= the initial value of -ip, ui = the angular velocity of OQ, (5 = the initial value of -p , then dt hi -pr - /B" ) = Saw" (cos \p - cos a). Clairaut; Mem. de VAcad. des Sciefices de Paris, 1736, p. 14. (9) QBA (fig. 145) is a circle on a horizontal plane, and QP a string toucliing it at the point Q; P is a particle attached to the end of the string. Supposing the particle P to be projected at right angles to QP with a given velocity so as to cause QP to be gradually wrapped about the cuxumference QBA ; to find the velocity of the particle at any time during the motion, and the time which will elapse before the particle reaches the circumference. Let /3 be the velocity of projection, c the velocity at any time during the motion, b the length of the string PQ, a the radius of the circle, T the time required. Then CONSTRAINED MOTION OF A PARTICLE. 281 (10) A circular horizontal lamina of matter ABC, (fig. 146), every particle of which attracts with a force varying inversely as the distance, is made to revolve with a uniform angidar velocity round an axis through its centre at right angles to its plane, the motion taking place in the direction of the arrows ; to find the equation to the groove Aa which must be carved in the cir- cular lamina that it may be described freely by a particle subject to the attraction of the lamina ; the initial position of the particle bemg a point A in the circumference of the circle, and its initial velocity being zero. Let P be any point in the groove, OP = r, OA = a, L POA = 0, (J) = the angular velocity of the lamina about O, and f = the at- traction of the lamina on a particle in its circumference. Then the equation to the groove Aa mil be (11) Two small equal bodies A, B, connected together by a rigid Hne, are placed in a narrow rectilinear tube, in which they can move without friction ; the tube is then made to revolve with a uniform angular velocity round a vertical axis which passes through a point C of the tube, this point C lying initially be- tween A and ^ at a distance a from A and h from B -, to find the tune of ^'s arriving at C, and the tension of the rigid line at any time, a being considered less than h. If u) denote the angular velocity of the tube, ni the mass of each particle, t the required time, and T the tension ; then ^ = ^log^^;, T=iW(a + J). - a (12) A particle is drawn up an indefinitely thin cycloidal tube, the axis of the cycloid being vertical, by means of an equal particle, to which the former particle is attached by a thi-ead passing over a pulley at the highest point of the arc ; to find the time of ascending to the highest point. If T represent the required time, and t the time of a semi- oscillation in the cycloid, i 282 CONSTRAINED MOTION OF A PARTICLE. (13) A heavy particle having been placed at a point in a straight line within a horizontal plane of indefinite length, round which as an axis the plane is then made to revolve downwards with a uniform angular velocity ; to find what time will elapse before the particle leaves the plane. If 0) be the angular velocity and t the required time, then 4 cos wt = s'"' + 8""". This problem was proposed in the Lady's Diary for the year 1 778, by the celebrated Landen, by whom a solution was given, which is singularly defective, not only in consequence of his neg- lecting the consideration of centrifugal force, but also from his erroneously considering the horizontal velocity of the particle to be equal to its velocity along the plane, midtiplied by the cosine of the plane's inclination to the horizon. See Diarian B,epository p. 512, where a correct solution is given by the Editors of the Repository, together with Landen's. Sect. 6. Constrained Motion of a Particle in Resistitiy Media. (1) A particle descends down a straight line AB, (fig. 147) inclined at an angle o to the vertical, in a medium of uniform density, in which the resistance varies as the velocity ; to deter- mine the velocity and the space at the end of any time. Let P be the position of the particle at the end of any time t, V its velocity, AP = x, and h = the resistance for a unit of velocity. Then, since the resolved part of the force of gravity along AB is at every point g cos a, we have for the motion of P, dv , -f - 9 cos a - kv, ^' = dt. g cos a - kv Integrating, we have C - y log {g cos a - kv) = t ; but « = when ^ = 0, and therefore C - y log (g cos a) = 0, CONSTRAINED MOTION OF A PARTICLE. hence 1 ^^ cos a - ^. _ k g cos a g cos a - kv _ _^^ g cos a ' g COS a ., V = ^-^— (1 - £-*^), which gives the velocity for any value of L Again, since dx = vdt, we have F/(-'-") ^ g cos a ,, , .,,. = (7+ - , a— (^"^+ £^): but, A being considered the initial position of the particle, „ g cos a hence x = S-^ (A^i - 1 i- i'"), which gives the position of the particle at any time. Euler; Mechan. torn. ii. p. 244. (2) A particle descends from rest by the action of gravity from a point E, (fig. 148), down the arc EA of a cycloid BAB' , of which the axis ^ C is vertical ; the motion takes place in a medium of uniform density, where the resistance is partly con- stant and partly proj)ortional to the square of the velocity; to find the velocity of the particle when it arrives at the point A, and to determine at what point in its descent its velocity is a maximum. Let AM= X, AP = s, AE= c, v = the velocity at P ; then, h and k being constant quantities, the equation of motion along the curve will be dv dx J If ds ds Ii ' hence d .v^ — v^ = - 2g dx + 2h ds : K 284 CONSTRAINED MOTION OF A PARTICLE. but, by the nature of the cycloid, if g a be the radius of the generatuig circle, dx = - ds', hence d .v^ — v^ = — - sds + 2hds. k a 2s Multiplying both sides of the equation by e *, we have 2s . 2s 2s d («- e" ^) = 2^6 ■* f/5 - ^ e' ^ 5^5. a Integrating, we have v^ ^^ = C-lik^'^ -M^"^ sds; ^'"Uds^-lk s" ^ 5 + U-J e'^ ds _2s 2.5 = -iks''' s-\k- e"^; 2s 2s J 2s hence v' e ' = C - hk s' ^ + - (gks + I gk') i ' , 1 -- = C+ - (gks + I g¥ - ahk) s *. But, initially, « = 0, s = c ; hence 1 - = C + - {gkc + ^ gk~ - ahk) g * . a Let Wj be the value of v when 5 = 0; then v'=C+-{\gk'-ahJi); k k -- and therefore v^ = -i^gk - ah) - -(gc + Igk- ah) e * . a d — I- - — 7- — Again, v" t'~=-{gs+lgk- ah) s ^ --{gc + \gk- ah) i \ a d' ^"^ = - {gs + \gk- ah) — {gc + ^gk- ah) e'' a ci When t5 is a maximum, , = g-*-3fe„+.gi-«A)J'n CONSTRAINED MOTION OF A PARTICLE. 28o (.-c) i ffk gc+lgk-ah' k , \gk k -, sc ■\- }i sh - ah 5 = C + - log ^T^ -, = C - - log 2 ^o_ 2 "^ gc + lgk-ah 2 ^ \gk which gives the position of the particle for a maximum velocity. Euler; Median, tom. it. p. 292. (3) From a given point 0, (fig. 1 49), an infinite number of straight lines OP are drawn in a vertical plane ; to determine the natui-e of the curve APD, such that a particle descending down any line OP may always acquii'e the same velocity on arriving at P, the medium in which the motion takes place being uniform, and its resistance varying as any power of the velocit5^ Let /3 be the velocity at P, v at any point p in OP ; OP = r, Op = z; L POx = B, Ox being vertical ; draw PM horizontally, and let OM=x; then, k being the resistance for a unit of velocity, and m the index of its power, V ---= g cos u - tiv , dz , vdv) dz — g cos - kv"" Integrating, we have r/^ vdv ^ p (3 d(5 J ^g cos B - kc"' J g cos - k(5"' ' But X = r cos ; hence Jog-k(5"'sec0 ^ ^ - '^' 1 z. a [^ i3"'^'^/3 .^. ~g- A/3"' sec ' '''^ '^^ '^ Jo (g - k[5- sec 0f"" ^''^^ But /3 being a constant quantity while x and vary, we have, from (1), dx = kd (sec 0) . (\ C'^^ ^,, ; and therefore, by (2), B~ in sec dx g - k j3'" sec d sec ' 286 CONSTRAINED MOTION OF A PARTICLE. or, putting - for sec B, m - dx + 2xcV ' ^ - k - j3" X and therefore (m - 2^ r dx + 2x dr = Q' — '■ j-r, — ; gx- ff(6"'r which is the differential equation to the curve in x and r. Euler ; Mechan. torn. ii. p. 246. (4) To find the tautochrone in a medium the resistance of which varies as the square of the velocity, the particle being acted on by gravity. Let (fig. 150) be the point to which the particle is always to descend in the same time, AO being the tautochrone. Take Oy horizontal as the axis of y, Ox vertical as the axis of x. Let OM = X, OP = s ; v = the velocity of the particle at P, and k = the resistance of the medium for a unit of a velocity. The equation for the motion along the curve will be vdv = - gdx + kv^ds : multiplying by 2e"^^', we have d (v\-^'') = - 2g5-''' dx. Integrating, we obtain v\-^^' = C- 2g fs--'' dx. Suppose the velocity of the particle on its arrival at to be that due to an altitude h in vacuo ; then 2gk = C- 2g fe-'"' dx; hence vV^''' = 2g {h - s'-'" dx} (1), Jo ds and therefore, v being equal to - — at any time t, (2g)' (h - uf where u = I e"^*' dx (2). J CONSTRATNED MOTION OF A PARTICLE. 287 Now, s being some function of z and therefore of u, we may assmne £"*' ds =

, in PM, p>'m, and construct the indefinitely smaU parallelogram P'pl , two of the sides of which are parallel to A C. Let AM = x, PM = y, P'M = y, Am = x + dx, p'ni = tj ^ dy' , lACB = C; a, h, c, the three sides of the triangle. Then, Mk^ denoting the moment of inertia about A, we have, T denoting the indefinitely small thickness, and p the density of the lamina, Mk^ = I (x^^ -h y'"^ - 'Ixy' cos C) pr sin C dx dy' J J = pr sin C I (x^y + ly^ - xy^ cos (7) dx J = OT sin O I I - x^ + \ —x^ - ^ x^ cos C ] dx Jo\b ^ F b"" J = ^.^T ab sin C (6b^ + 2a^ - 6ab cos C) = i^M {6b' + 2a' - 3 (a' + b' - c')}, and therefore k' =^, (3b' + 3c' - a'). (2) To find the radius of gyration of a triangular lamina ABC ahoxit a perpendicular through its centre of gravity G. Let AG, BG, CG, be represented by a, j3, 7 ; and BC, AC, AB, by a, b, c. Then, 31 denoting the mass of the whole 296 MOMENT OF INERTIA. triangle ABC, k M will be the mass of each of the triangles BGC, AGO, AGB. Hence, by the preceding problem, the moment of inertia of these thi-ee triangles respectively about the axis through G "will be and therefore the moment of inertia of the whole triangle about G will be equal to i,3I{6 (a- + j3^ + y) - (a' + b' + c')} ; or, by a property of the centre of gravity of a triangle, to ^M{2 (a- + 6* + c') - (a' + b' + c')] = ^ M{a' + b' + c'). Hence the radius of gyration will be equal to Euler; Theoria Motus Corporum Solidorum, cap. vt. Prob. 32. Cor. 1. (3) To find the radius of gyration of an elHptic area about a perpendicular axis through its centre. If M be the mass of the area, the moment of inertia about the two axes of the ellipse will be \Mb\ \Ma\ But the moment of inertia of a plane area, about any perpen- dicular axis, is equal to the sum of the moments of inertia about any two lines, at right angles to each other in the plane area, passing through the point in which the axis meets the area. Hence, in the present problem, the moment of inertia about the proposed axis is equal to i M{d- + b% and the radius of gyration = \ (cr + W). (4) To find the radius of gp-ation of an annulus about a per- pendicular axis thi'ough the centre. Let r be the distance of any point of the annular area from the centre of the circle, d the angular co-ordinate, p the density, and MOMENT OF INERTIA. 297 T the indefinitely small thickness of the area ; then, a, h, being the radii of the two concentric circles, /•27r /"ft Mk^ = r. prrdSdr Jo J a = \pr\ (i* - a') dB = i irpr (h' - a*). J /'21- rb /•2T But M=\ \ prrdOdr = ^, pr Qr - a^) dS = TTjor (b~ - a^) ; hence F = i («' + *')• (5) To find the radius of gyration of a parallelogram about an axis perpendicular to it through its centre of gravity. If 2a, 2b, be the lengths of two adjoining sides of the parallelo- gram, then, whatever be the angle of their inclination, Euler ; Theoria Motus Corp. Solid, cap. vi. Prob. 35. (6) To find the radius of gyration of a regular polygon about an axis perpendicular to it through the centre. If n be the number of sides, and c the length of each, 2ir 2 + cos ^2 12 F = ' C' "" 27r 1 - cos ?l (7) To find the radius of gyration of a portion of a parabola bounded by a double ordinate to the axis about a perpendicular line through its vertex. If X, y, represent the extreme co-ordinates of the portion. Sect. 5. Symmetrical Solid about its Axis. (1) To find the radius of gyration of a homogeneous sphere about a diameter. Let X, X + dx, be the distances of the cu'cular faces of a thin circular slice of the sphere, at right angles to the diameter, from 298 MOMENT OF INERTIA. the centre, and let y be the radius of the section. Then, p denoting the density of the sphere, the moment of inertia of this sHce about the diameter will be equal to \ irpy^dx ; and therefore the moment of inertia of the whole sphere, a being its radius, will be equal to 5 TTp I y^dz = \irp\ (a^ - x'J dx ^ fs "Tpa". But the mass of the sphere is equal to 3 Trpa^ ; hence Euler; Theoria Motiis Corporum Solidorum,-p. 198. (2) To find the radius of gyration of a right cone about its axis. If a denote the radius of the base of the cone, A/ — ^g O . Euler; lb. p. 197. (3) To find the radius of gyration of a hollow sphere about a diameter. If a, h, be the external and mternal radii, 72 _ o a" - b" Euler; lb. p. 203. (4) To find the radius of gyration of a solid cylinder about its axis. If a denote the radius of the cylinder, k'^ = i a\ Euler; lb. p. 200. Sect. 6. Moment of Inertia of a Solid not Symmetrical with respect to the Axis of Gyration. (1) To find the radius of gyration of a soHd cylinder about an axis perpendicular to its own through its middle point. Let X be the distance of any thin circular sHce of the cylinder from the middle point of its axis ; c^ the thickness of the sKce ; MOMENT OF INERTIA. 299 p the density of the cylinder, b its radius, and 2a its length. Then, the moment of inertia of the slice about any diameter being equal to 5 Trph*dx, its moment of inertia about the axis of gyration of the present problem will be equal to irplrdx .{x^ + \ F). Hence, MJ^ denoting the moment of inertia of the whole cylinder about the proposed axis, we have MJ^ = TTph" r {x' + i b^) dx = irpV (i a' + ^ ab^) = 27rpaPQia' + \b^); and therefore, M being equal to ^irpah^, we have ^^ = i a^ + i b\ Euler; Theoria Motus Corporum Solidorum, p. 196. (2) To find the radius of gyration of a right cone about an axis at right angles to the axis of the cone and passing through its centre of gravity. If a be the altitude of the cone, and c the radius of its base ; then k' = ^ (a^ + 4c^). Euler; lb. -p. 197. (3) To find the radius of gyration of a right cone about an axis through its vertex at right angles to its geometrical axis. If a = the altitude of the cone, and c = the radius of the base, (4) To find the radius of gyration of a double convex lens about its axis, and about a diameter to the circle in which its two spherical surfaces intersect ; the two surfaces having equal radii. If a = the semi-axis of the lens, b = the radius of the circidar intersection of the two surfaces ; k = the radius of gyration of the lens about its axis, and k' about a diameter of the circle ; we shall have , a* + 5a:b- + lOb^ ,,2 j 7a*+ 15a'6'+10Z»' K — -on ^ a'+3b' ' ~'' ^f7W ■ Euler; lb. p. 201. ( 300 ) CHAPTER VI. D ALEMBERT S PRINCIPLE. A GENERAL method for the determination of the motion of a material system, acted on by any forces, was laid down by D'Alembert in his Traite de Dynamique, published in the year 1743^, from which we have extracted the following passage in exposition of the Principle.' " ProhUme General. " Soit donne un systeme de corps disposes les uns par rapport aux autres d'une maniere quelconque; et supposons qn'on im- prime a chacun de ces corps un mouvement particulier, qu' il ne puisse suivre a cause de 1' action des autres corps ; trouver le mouvement que chaque corps doit prendre. " Solution. " Soient A, B, C, Sec. les corps qui composent le systeme, et supposons qu' on leur ait imprime les mouvemens a, h, c, etc. qu' lis soient forces, a cause de leur action mutuelle, de changer dans les mouvemens a, b, c, etc. II est clair qu'on pent regarder le mouvement a imprime au corps A comme compose du mouve- ment a, qu' il a pris, et d' un autre mouvement a ; qu' on pent de meme regarder les mouvemens h, c, etc. comme composes des mouvemens b, j3 ; c, k', etc. d' ou il s' ensuit que le mouvement des corps A, B, C, etc. entr' eux auroit ete le meme, si au lieu de leur donner les impulsions a, h, c, on leur eut donn^ a-la-fois les doubles impulsions a, a ; b, j3 ; c, k, etc. Or par la supposi- tion, les corps A, B, C, etc. ont pris d' eux-memes les mouve- mens a, b, c ; etc. done les mouvemens a, /3, k, etc. doivent etre tels qu' ils ne derangent rien dans les mouvemens a, b, c, etc. ' See also his Recherches sur la Precession des Equinoxes, p. 35, published in 1740. ^ D'Alemberi's Principle was first enunciated by him in naemoir which he read before the Academy of Sciences at the end of the year 42. d'alembert's principle. 301 c' est-a-dire que, si les corps n' avoient re9u que les mouvemens a, j3, K, etc. ces mouvemens auroient du se detruire mutuelle- ment, et le systeme demeurer en repos. ^~~" De la resulte le principe suivant, pour trouver le mouvement de plusieurs corps qui" agissent les uns sur les autres. Decom- posez les moutemens a, h, c, etc. imprimes a chaque corps, cha'cun en deux autres a, a ; b, j3 ; c, k; etc. qui soient tels, que si r on rC eut imprime aux corps que les mouvemens a, b, c, etc. ils eussent pu conserver ces moutemens sans senuire reciproquement ; et que si on ne lew eut imprime que les moutemens a, fi, k, etc. le systeme fut demeure en repos ; il est clair que a, b, c, seront les mouvemens que ces corps proidront en vertu de leur action. Ce qu^ ilfalloit trouver.''^ The idea of the general method developed by D'Alembert for the determination of the motion of material- systems, had occiu'- red somewhat earlier to Fontaine, as w^ are informed in the Table des Memoires, prefixed to his Traite de Calcul Differentiel et Integral^ having been communicated by him to the Academy of Sciences in the year 1739, and subsequently to several mathema- ticians. His views, however, on this subject were not made public till long after the appearance of the Traite de Dynamique; and in all probability D'Alembert, who did not become a member of the Academy before the year 1741, was not aware of Fontaine's generalization. D'Alembert, however, was the first to shew the wonderful fertiHty of the Principle by applying it to the solution of a great variety of difficult problems, among which may be mentioned that of the Precession of the Equinoxes, which had been inadequately attempted by Newton, and of which D'Alem- bert was the first to obtain a complete solution. The earliest step towards the discovery of D'Alembert's Prin- ciple is to be met with in a memoir by James Bernoulh in the Acta Eruditorum, 1686, Jul. p. 356, entitled " Narratio Contro- versiae inter Dn. Hugenium et Abbatem Catelanum agitatae de Centro Oscillationis quae loco animadversionis esse poterit in Eesponsionem Dn. Catelani, num. 27. Ephem. Gallic, anni 1684, insertam." Let m, m' , denote two equal bodies attached to an ' Memoires de V Acudemie des Sciences de Paris, 1770. 302 d'alembert's principle. inflexible straight line which is capable of motion in a vertical plane about one extremity which is fixed ; let r, r', denote the distances of m, ?n', respectively, from the fixed extremity ; v, v', then- velocities for any position of the inflexible line in its descent ffom an assigned position, ii, u; the velocities which they would have acquired by descending down the same arcs unconnectedly. Then, in consequence of the connection of the bodies, a velocity u - V will be lost b)^ ?w and a velocity v' - u gained by 7n' in then- descent. Bernoulli proposes it to the consideration of mathema- ticians whether, according to the statical relation of two forces in equilibrium on a lever, the proportion u - v : v - ti :: r : r be an accurate expression of the cuxumstances of the motion. This idea of Bernoulli's, although not free from error, contains how- ever the fii'st germ of the Principle of reducing the determination of the motions of material systems to the solution of statical problems. L'Hopital, in a letter addressed to Huyghens,^ cor- rectly observed that instead of considering the velocities acquired in a finite time, he should have considered the infinitesimal velo- cities acquired in an instant of time, and have compared them with those which gra^dty tends to impress upon the bodies diu'ing the same instant. He takes a complex pendulum, consisting of any two bodies attached to an inflexible straight line, and considers equilibrium to subsist between the quantities of motion lost and gained by these bodies in any instant of time, that is, between the differences of the quantities of motion which the bodies really acquire in this instant, and those which gravity tends to impress on them. He applies this Principle, which agi-ees with the general Principle of D'Alembert, to the determination of the Centre of Oscillation of a pendulum consisting of two bodies at- tached to an inflexible straight line oscillating about one extre- mity. He then extends his theory to a greater number of bodies in a straight line, and determines their Centre of Oscilla- tion on the supposition, the truth of which is not however sufficiently obvious M'ithout demonstration, that any two of them may be collected at their particular Centre of Oscillation. On the pubHcation of L'Hopital's letter, James Bernoulli" reverted ' Histoire cles Ouvrages des Sqavans, 1690, Jain. p. 44'0. * Acta Enidit. Lips. 1691. Jul. p. 317. Opera, torn. I. p. ■ieO. d'alembert's principle. .303 to the subject of the Centre of Oscillation, and at length suc- ceeded in obtaining a direct and rigorous solution of the problem in the case where all the bodies are in one line, by the appli- cation of the principle laid down by L'Hopital. Bernoulli' afterwards extended his method to the general case of the oscil- lations of bodies of any figure. An ingenious investigation of the Centre of Oscillation, a problem from the beginning intimately connected with the deve- lopment of D'Alembert's Principle, was shortly afterwards given by Brook Taylor' and John Bernoulli,^ between whom arose an angry controversy respecting priority of discovery ;* the method given by these mathematicians, although depending upon the statical principles of the lever, did not however involve, in an explicit form, L'Hopital's Principle of Equilibrium. Finally, Hermann* determined the Centre of Oscillation by the principle of the statical equivalence of the solicitations of gravity, and the vicarious solicitations applied in opposite dii'ections, or, as it is expressed by modern mathematicians, by the equilibrium subsist- ing between the impressed forces of gravity and the effective forces applied in opposite directions ; a method of investigation virtually coincident with that given by James Bernoulli. The idea of L'Hopital became still more general in the hands of Euler,^ in a memoir on the determination of the oscillations of flexible strings printed in the year 1740. From the above his- torical sketch it will be easily seen that in the enunciation of a general Principle of Motion, Fontaine and D'Alembert had little more to do than to express in general language what had been distinctly conceived in the prosecution of particular re- searches by L'Hopital, James and John Bernoulli, Brook Taylor, Hermann, and Euler. For additional information on the histori- cal development of D'Alembert's Principle, the reader is referred ' Memoires de VAcademie des Sciences de Paris, 1703, 1704. * Philosophical Transactions, 1714', May. Methodus Incremeniorum. * Acta Erudit. Lips. 1714. Jun. p. 257; Mem. Acad. Par. 1714, p. 208. Opera, torn. ir. p. 168. ' Act. Erudit. Lips. 1716, 1718, 1719, 1721, 1722. ^ Phoronomia ; lib. I. cap. 5. " Comment. Petrop. torn. vn. 304 d'alembert's principle. to Lagrange's M^canique Analytique, Seconde Partie, Section 1 ; Montucla's Histoire des Mathhnatiques , part. v. liv. 3, part. iv. liv. 7 ; and Whewell's History of the Inductive Sciences, vol. ii. In modern treatises on Mechanics, D'Alembert's Principle is expressed under one or other of the following forms : (1) When any material system is in motion under the action of any forces, the moving forces lost by the different molecules of the system must be in equilibrium. (2) If the effective moving forces of the several particles of a system be applied to them in directions opposite to those in which they act; they will, conjointly with the impressed moving forces, constitute a system of forces statically disposed. The former of these enunciations it will be seen is substantially the same as that given by D'Alembert, while the latter is a generalization of the idea developed by Hermann in his iavesti- gations on the particular problem of the Centre of Oscillation. Sect. 1. Motion of a single Particle. The object of this section is to apply D'Alembert's Principle to the exemplification of a general method for the determination of the motion of a particle within tubes and between contiguous surfaces, of which either the position, or the form, or both, are made to vary according to any assigned law whatever, the particle being acted on by given forces. Several of the problems of this section have been solved by particular methods in Chapter iv. I. "We will commence with the consideration of the mo- tion of a particle along a tube, and, for the sake of perfect generality, we will suppose the tube to be one of double curva- ture. The tube is considered in all cases to be indefinitely narrow and perfectly smooth, and every section at right angles to its axis to be circular. Let the particle be referred to three fixed rectangular axes, and let x, y, z, be its co-ordinates at any time t ; let x, y, z, be- come X + ^x, y + Sy, z + dz, when t becomes t + Bt; St, and ' The substance of this Section was published in the Cambridge Mathematical Journal, vol. in. p. 49. d'alembert's principle. o05 consequently Bx, Sy, Sz, being considered to be indefinitely small. Then the effective accelerating forces on the particle parallel to the three fixed axes will be, at the time t, S'x ^ ^ . If ' se ' sf ' Also, let X, Y, Z, represent the impressed accelerating forces on the particle resolved parallel to the axes of x, y, z; and let X + dx, y + dy, z + dz, be the co-ordinates of a point in the tube very near to the point x, y, z, which the particle occupies at the time t. Then, observing that the action of the tube on the particle is always at right angles to its axis at every point and therefore, at the time t, to the line joining the two points x, y, z, and X + dx, y + dy, z + dz, we have, by D'Alembert's Principle, combined with the Principle of Virtual Velocities, Again, since the form and position of the tube are supposed to vary according to some assigned law, it is clear that when t is known the equations to the tube must be known ; hence it is evident that, in addition to the equation (A), we shall have, from the particular conditions of each individual problem, a number of equations equivalent to two of the form ^ {x, y, z, 0=0, X (^' y? ^' = (B), where and ^ are symbols of functionality depending upon the law of the variations of the form and position of the tube. The tliree equations (A) and (B) involve the four quantities ^> y^ ^} i} ^nd therefore, in any particular case, if the difl[iculty of the analytical processes be not insuperable, we may ascertain X, y, z, each of them in terms of ^ ; in which consists the com- plete solution of the problem. If the tube remain during the whole of the motion within one plane, then, the plane of x, y, being so chosen as to coincide with this plane, the three equations (A) and (B) will evidently be reduced to the two (g_x)<;. + (&-r)rfy = o (C), ^Cx,i/,t)=0 (D). 306 d'alem Bert's prixciple. We proceed to illiistrate the general formulae of the motion by the discussion of a few problems. (1) A rectilinear tube revolves with a uniform angular velocity about one exti'emity in a horizontal plane ; to find the motion of a particle within the tube. Let w be the constant angular velocity ; r the distance of the particle at any time t from the fixed extremity of the tube ; then the plane of x, y, being taken horizontal, and the origin of co-ordinates at the fixed exti'emitj^ of the tube, we shall have, supposing the tube initially to coincide with the axis of z, X = r cos tot (1), y = r sin w^ (2). From (1) we have dz = dr cos wf, and from (2), dy = dr sin (»t. Again, fi-om (1) we have ^z h- ^ . ^ K- - -^ cos wt - wT sm wf, ^'x SV , ^ ^r . ^ , -K2i = j7i cos b)t - 2w -^ sm wf - w r cos wt ; and fi'om (2), ^y Br . ^ ^ = -^ sm ujt + wr cos wf. Of of B'y ^r . ^ ^ Sr ^ ■> • ^ ■=^ = =-2 sm wf + 2w Y" cos wf - w'r sm wr. ctr ot ct Substituting in the general formula (C) the values which we have obtained for dz, dy, ^^ ,~ , we have, since X = 0, Y= 0, O^ bt' cos i»t ( -K-j cos wt - li,) Y sm o)t - to r cos wt + sm wt TT^ sm wf -f 2w rr- COS wt - w r sm wt \ct- ht ) = 0; and therefore ^V d'alembert's principle. 307 the integral of this equation is r = a"" + C'f-' Let r = a when ^ = ; then a= C+ C; also let -s;- == j3 when ^ = ; then 0^ ^ = Cu) —Cm', from the two equations for ■ the determination of C and 6", we have C = aw + ii ^, _ aw -/3. 2w 2t hence for the motion of the particle along the tube 2u>r = (aio + j3) s'"' + (aio - j3) e"'"'. This problem, which is the earliest problem of the motion of a particle subject to the constraint of a curve moving according to a prescribed law, is due to John Bernoulli^ A solution of this problem is given also by Clairaut", to whom it had probably been proposed by Bernoulli. (2) Supposing the tube to revolve in a vertical instead of a horizontal plane, we shall have, by the same process, the axis of y being now taken vertical, observing that X = 0, Y =- g sin wt, if the time be reckoned from the moment of coincidence of the tube with the axis of x which is horizontal, ^ - w r = - ^ sm w^. The integral of this equation is r = Ce'"* + C't-"'' + -^, sin wt; 2w^ Ir and if we determine the constants from the conditions that r, -^ , shall have initially values a, j3, we shall get for the motion along the tube, 2wr = iciM + /3 - -^ J E*"' + ( aw - j3 + -^"j e"" + t sin i^t. ' Opera, torn. iv. p. 248. ' Memoires de I'Academie des Sciences de Paris, 1742, p. 10. x2 308 d'alembert's principle. This problem, which had been erroneously attempted by Barbier in the Annales de Gergomie, tom. xix., was correctly solved, in the following volume, by Ampere. In the Cambridge Mathematical Journal, vol. iii. p. 42, a solution is given by Professor Booth, who has discussed at length the more interest- ing cases of the motion. (3) Suppose the tube to revolve in a horizontal plane about a fixed extremity with such an angular velocity, that the tangent of its angle of inclination to the axis of x is proportional to the time. The equation to the tube at any time t will be y = mtx (1), where m, is some constant quantity ; hence dy = mt dx, and therefore from (C), since X= and Y = 0, ^^X ^ SV r. /r.\ But from (1) we have hence, from (2), ^y ^ Bx ^=^mt^^mx, ^=mt^^2m^; ^x Sf 2mH St Integrating, we have log|? + log(l+m¥) = logC, Sx . . Let /3 be the initial value of -=^ , which will be the velocity of d'alembert's principle. 309 projection along the tube; then C= (5, and therefore St ^ ^ ^' ' 1 + m-f integrating, Are get X + C = — tan"' (mf). m Let X = a when ^ = ; then a + C = 0, and therefore Q X = a + — tan"' (^Oj m and consequently, from (1), y = ami + j3^ tan"' {mf). If be the inclination of the tube to the axis of x at any time, and r be the distance of the particle from the fixed extremity, ^ _ am + j30 m, cos B (4) A circular tube is constrained to move in a horizontal plane with a uniform angular velocity about a fixed point in its cu'cumference ; to determine the motion of a particle within the tube, which is placed initially in the extremity of the diameter passing tkrough the fixed point. Let the fixed point be taken as the origin of co-ordinates, and let the axis of x coincide with the initial position of the diameter thi-ough this point ; let w be the angular velocity of the revolu- tion of the circle, a the radius; also let be the angle at any time t between the distance of the particle and of the extremity of the diameter tlii-ough the origin from the centre of the cii'cle. Then it will be easily seen that X = a cos b)t + a cos (w^ - &) (1), y = a sin wt -v a sin {wt - 0) (2). From (1) we have dx = add sin (wt - 0), and, from (2), dy = - adO cos (wt - 0). Hence, from (C), observing that X = and Y = 0, sin {a -0)^- cos (c.^ - 0) i| := (3). o^' of ~- = au) cos o) St 310 d'alembert's principle. Again, from (1), -»- = - aw sm ot + a { ^ w J sin (wr - u), — = - aw" cos w^ - a ( r^ w j cos (w^ - a) + a ^ sm (wf - (/; ; and, from (2), t - a (-^ — w| cos (b)i - 0), ■^ = - aw sm w# - a -^ — w 1 sm (w^ - b) - a -^ cos (wr - u) ; and therefore, by (3), aw^ {sin u)t cos (w^ - 0) - cos wt sin (w? - ^)} + ^ -s;;^ = ^? or" w^ sin + ^ = : multiplying by 2 — , and integrating, ■^ = C + 2w^ cos 9. But, the absolute velocity of the particle being initially zero, it is clear that 2w will be the initial value of -s^ ; and therefore, 9 ot being initially zero, we have 4w' = C + 2w^ C = 2w^ and therefore S9' , , ^, 2 2 S0 . -^ = 2w- (1 + cos C*) = 4w cos - , -^ = 2w cos - , cos - ^9 S sin - ...^=2w8^, l = .dt. 2 " , • 2 " cos - 1 - sm - 2 2 Integrating, w^e have 1 • ^ 1 + sm - 2 log = 2w^ + C; 1 - sm - 2 d'alembert's principle. 311 but 6=0 when t = ; hence C = 0, and we have 1 + Sin - 1 - sm — 2 and therefore sm - = 2 £"-' + £-<"' " which determines the position of the particle within the tube at a any time. When t = oo , we have sin - = 1 , and therefore = ir, which shews that^ after the lapse of an infinite time, the particle will arrive at the point of rotation. (5) If we pursue the same course as in the solution of the problems (1), (2), (4), we may obtain a convenient formula for the following more general problem : a plane curvilinear tube of any invariable form whatever revolves in its own plane about a fixed point with a uniform angular velocity ; to determine the motion of a particle, acted on by any forces, within the tube. Let w be the constant angular velocity of the tube about the fixed point ; r the distance of the particle at any time from this point; (j) the angle between the simultaneous dii-ections of r and of a line joining an assigned point of the tube with the fixed point of rotation ; ds an element of the length of the tube at the place of the particle, and *S'the accelerating force on the particle re- solved along the element ds ; then the equation for the motion of the particle will be but since, the form of the tube being invariable, S^, Sr, may evidently be replaced by d(f), dr, we have, putting, for the sake of uniformity of notation, dt in place of St, If w be zero, the formula will become /■ 312 D AT,EMBERT S PRINCIPLE. the well-known formula for the motion of a particle under the action of any forces within an immoveable plane tube. (6) In the foregoing examples the position of the tube varies with the time, the form however remains invariable. "VVe will now give an example in which the form changes with the time. A particle is projected with a given velocity within a circular tube, the radius of which increases in proportion to the time while the centre remains stationary ; to determine the motion of the particle, the tube being supposed to lie always in a horizontal plane. The equation to the circle will be x^ + ?/- = a'- (1 + off (1), where a and a are some constant quantities ; hence X dx + y cly = 0, and therefore, by the general formula (C), integrating, we have ^'x gV ^ ^ St St Let the axis of x be so chosen as to coincide with the initial distance of the particle from the centre, and let j3 be the initial velocity of the particle along the tube ; then C = - a/3, and therefore By Bx ^ .^ again, from (1), we have ^| + y| = «^«(l+a0...(3); multiplying (2) by y and (3) by x, and subtracting the former result from the latter, we have Bx (^' + y') g^ = «'« (1 + °^) ^ - ^^y^ and therefore, by (1), Bx St Sx i a (I -f utf -^ = aa(l + nt) X - /3 {a- (1 + atf - x-y. d'alembert's principle. 313 Put 1 + a^ = T ; then aar ^ = aarX - (5 (a'r - X^J ; OT again, put x = nir, and there is Clar [m + r tt" ] = aanir - pr{a - m) , integrating, aar -^ = - /3t («- - m ) , or - aa _ = f3 — ; (a - m ) ri -1 ^^^ ^ C + aa cos — = - - , a or, putting for wi its value, ri -I ••^ /^ C + aa cos — = - - , ar T and putting for t its value 1 + at, C+ aa cos = ,. a (1 -t a^) • \ + at Now a; = rt when ^ = ; hence C = - (5, and therefore iP a(St aa COS a (1 + a^) 1 ^ at' Bt X = a(l + at) COS —yi- , a{l + at) and therefore, from (1), which give the absolute position of the particle at any assigned time. II. We proceed now to the consideration of the motion of a particle along a surface from which it is unable to detach itself, while the surface itself changes its position or its form, or both, according to any assigned law. To fix the ideas, we suppose the particle to move between two surfaces indefinitely close together, so as to be expressed by the same equation. 314 d'alembert's trinciple. Let X, y, z, be the co-ordinates of the particle at any time t ; and let Sx, Sy, Sz, be the increments of x, y, z, in an indefinitely small time ^t ; also let dx, dy, dz, denote the increments of x, y, z, in passing from the point x, y, z, to any point near to it within the surface as it exists at the time t. Also let X, Y, Z, denote the resolved parts of the accelerating forces on the particle at the time t parallel to the axes of x, y, z; then, observing that the action of the surface on the particle is always in the direction of the normal at each point, we have, bv D'Alembert's Principle combined with the Principle of Virtual Velocities, Again, since the position and form of the surface vary according to an assigned law, its equation must evidently be known at any given time, and therefore we must have, from the nature of each particular problem, certain conditions between the quantities x, y, z, t, equivalent to a single equation F=f{x,y,z,t)^Q (B'). Taking the total differential of {B'), we have dF , dF , dF , ^ -r~ dx + -^- dy + ^- dz = ; dx dy dz eliminating dz between this equation and (^'), we get ^'x ^\ dF , /gV ,A dF , ^'z y\fdF, dF, but dx and dy are independent quantities ; we have therefore, by equating separately their coefficients on each side of the equation, w )~d^' \Je J dx ' ( ^Se J dz \Sf J dy ' and therefore also fS^ \dF f^_ \ W~^ J dy'\Bf~ J (IF dx d'alembert's principle. 315 any tAvo of these three relations, together with the equation {E), will give us three equations in x, y, z, t, whence x, y, z, are to be determined in terms of t. The following example will serve to illustrate the vise of these equations. AVe have taken a case where the form of the surface remains invariable, its position alone being liable to change. The analysis, however, in the solution of problems of the class which we are considering, receives its general character solely in consequence of the presence of t in the equation {B), and therefore the example which we have chosen is sufficient for the general object we have in view. A particle descends by the action of gravity down a plane which revolves uniformly about a vertical axis through which it passes ; to determine the motion of the particle. Let the plane of x, y, be taken horizontal, the axis of x coin- ciding ■svith the initial intersection of the revolving plane with the horizontal plane through the origin, and let the axis of z be taken vertically downwards ; then, w denoting the angular velo- city of the plane, its equation at any time t will be F = y cos w^ - a; sin w^ = (1); whence ^p . dF , clF -^- = - sni h)t, —f- = cos (i)t, — - = ; ax ay dz also X = 0, Y= 0, Z = g ; and therefore, from either of the two first of the three general relations, w'" ('^' and from the third, _ cos w^ + g^ sm w/ = (3). Let r denote the distance of the particle at any time from the axis of z ; then x = r cos ii)t, y = T sin wt, , & gr ^ whence -;r- = k- cos lot - wr sm lot, St St S^x S-r / -, ^'" • / ' ir-5 = -;r-; cos (ot - 2w TT- Sni (ot - W /' COS wf, Sf Sf St 316 D ALEMBERT S PUINCIPLE. By Br . ^ ^ = »- sm u)t + wr cos tjt, bo ct By B'r . ^ ^ Br ^ " • ^ ^ =^ K^ sm wf + 2w =^ cos ot - io'r sin ut ; Of" or" o^ and therefore, from (3), !?-"''■=" «• Bz Br Let the initial values of 2, ^ , be 0, /3 ; and those of r, ^ , hi of be a, a ; then, from the equations (2) and (4), after executing obvious operations, we shall obtain z = lgf + (it, 2wr = (loa + a) t'"' + (wa - a) e''^% 1 J , (w~r~ + a — b) a)' + o)r '^ ur, o2\l oi and log ' — — ^ = - {(2gz + (i~) - (i] ; a + wtt i^d' + im ^^' Let T denote the tension of the string supporting m, then rr I CI X 1 = m[q :rv, V dt' ( d'O = m[ q - a — — V^ df _ { -, (i (^^ - m!a') 1 ~ ^^ \ ~ ma^ + m'a" + Mk^] _ ma {a + «') + MH Similarly, the tension of the other string being denoted by J", , ma {a + a) + MTi ^'^^ m'a" + md+ Mk'' (4) A uniform heavy rod OA, (fig. 158), which is at liberty to oscillate in a vertical plane about a horizontal axis through O, falls from a horizontal position ; to determine the angle included between the direction of the rod and the direction of the pres- sure for any position of the rod. Let Ox, Oy, be the axes of co-ordinates in the plane of oscilla- tion, Ox being horizontal and Oij vertical ; let Oz be at right angles to the plane xOij. Let U, V, represent the resolved parts of the reaction of the axis Oz upon the rod, estimated along xO, 320 d'aLEMBERT's PRINCTPLE. yO. Let p = the density of the rod, k = the area of a section of it taken at right angles to its length ; let P be any point in OA, di"aw PM at right angles to Ox ; let 0M= X, PM= y, OP = r, OA = a, lAOx = 0. Then, by D'Alembert's Principle, resolving forces parallel to Ox, 7-/:f'"^^f}=--/:(*§) '^^■' resolving forces parallel to Oy, and, taking moments about the axis Oz, But, from the geometry, we see that ^ clx . ^ dO (fx n fl^~ • /> d'Q X = r cos if, -r- = - r sin (/ -— , -— = - r cos t^ — ^ - r sm t; ^-, , dt dt df df df ' and similarly hence, fr'om (1), we have d'y . ^dff- ^d'B _ = -.sm0^ + rcos0-,; Z7= Kp I rdr [ cos ^ + sin dO' . ^cf-e' \ dt dt^ = ia.p^cos0^ + sin0^J ^4)' and, from (2), V= Kpga + I a'Kp [ sin -^ - cos — - J (5). Again, from (3), substituting for x and y their values in terms of r and B, we get g \ cos rdr = I rV/r -y^ , Jo Jo dir and therefore hqa cos B = \ar ^^ , -^^ = -^ cos B; "^ ^ df df 2a d'alembert's principle. 3^1 multiplying by 2 — , integrating, and bearing in mind that B = Q when — - = 0, we have dt ' — = -^ sin e. dr a dfi rPR Hence, substituting for — and -j-^ their values in (4) and (5), we obtain U=^Kp ag sin 9 cos 0, F= iK:pa<7(10 - 9 cos'0). From these equations we get Ucos B + F" sin = I (cp «^ sin B, V cos B - ?7 sin = 5 icp ag cos B. But U cos B + Vsin B and V cos B - U sin B are the expres- sions for the resolved parts of the reaction of the fixed axis, estimated along AO and at right angles to ^0 ; hence, if denote the inclination of the resultant reaction to the line AO produced, or of the resultant pressure on the axis to the line OA, we shall have rcos^-^Jsina , tan ^ = _ /) - T^ • n = TO cot B, (7 cos B + Vsm B tan B tan lo,num, but also in certain instances of oscillation in latus, while together with Descartes he foiled to give a correct solution of the problem in the case of solid figures. The labours of Huyghens, who in his earlier efforts to obtain a solution of Mersenne's problem had been utterly baffled, were at length crowned with success, and accordingly in the fourth part of his Horologium Oscillatorium, which appeared in the year 1673, was given the first rigorous and general investigation of the Centre of Oscillation. The two folloAving axioms constitute the basis of his researches : first, that the centre of gravity of a system of heavy bodies cannot of itself rise to an altitude greater than that from which it has fallen, whatever change be made in the mutual disposition of the bodies ; and secondly, that a compoimd pendulum will always ascend to the same height as that from which it has descended freely. Some years after the publication of the Horologium Oscillatorium, the truth of these fundamental axioms, which although true, it must be admitted, are not suffi- ciently elementary, was called in question by the Abbe Catelan,^ who substituted certain frail theories of his own in place of the valuable researches of Huyghens. The attention of the mathe- maticians of the day having been more closely directed to the subject by the controversy which arose between Huyghens and Catelan, the views of Huyghens received ample corroboration from the more elementary investigations of L' Hopital, James Bernoulli, and other mathematicians. For information respect- ing the subsequent history of Mersenne's problem, the reader is referred to the Chapter on D'Alembert's Principle. (1) To find at what point of the rod of a perfect pendulum must be fixed a given weight of indefinitely small volume, so as to have the greatest effect in accelerating the pendulum. Let m be the mass of the bob of the perfect pendulum, and a its length ; iri the mass of the given weight, and a the distance of its point of attachment from the centre of suspension ; / the distance between the centre of suspension and the centre of ' Journal des Scavuii.s, 1()82 ul 1(j81. 326 MOTION OF RIGID BODIES ABOUT FIXED AXES. oscillation of the complex pendulum. Then we shall have, m and m being both of indefinitely small volume, 7na + ma Now the shorter the rod of a perfect pendulum, the shorter will be the time of its oscillations ; hence we must have / a minimum ; differentiating then with respect to a we get (II _ 2m'a' {ma + ma) - m' (ma' + m'a''^) da (tna + m'a'f hence m'a'^ + 2maa' = ma", m'^a'^ -I- 2ma m'a + m~a' = (m' + mm') a", i m'a -f ma - {m^ + vwija, a = — { (m + mm ) - m] , ml J which determines the required point of attachment. Lady^s and Gentleman^ s Diary, 1742. Diarian lieposi- toty, p. 394. Euler; De Ilotu Corp. Solid., Prob. 48. Cor. 1. p. 216. (2) To compare the times in which a circular plate will vibrate round a horizontal tangent and round a horizontal axis, through the point of contact, at right angles to the tangent. Let /, /', denote the lengths of the isochronous pendulums in the former and latter case respectively ; a the radius of the plate ; k, Ji , the radii of gyration about axes tlu'ough the centre of the plate parallel in each case to the axis of oscillation. Then , «^ + /*;" J, a' + k'^ a a Let A denote the area of the plate ; r the distance of a point within it from its centre, and 6 the inclination of this distance to the horizon when the plate is hanging at rest. Then Ak' = fJrdOdr . r' sin^ 9, between the proper limits. i' fr' siirO dddr = ] a'f' ^hvedO J 11 J Jo MOTION OF RIGID BODIES ABOUT FIXED AXES. C Also Ak" = 1 1 rdOdr .r'=j J r'dOdr = i «" j dO = h Tra\ Bat ^ = 7ra" ; hence k' = ^d^ and k'^ = 5 a^ ; and therefore l = a + \a = ^a, I' = a + \ a = ^a. Hence, if t, t' , denote the thnes of vibration, (3) If / and h be the distances of the centres of oscillation and gravity of a mercurial pendulum of which the weight is m, from the axis of suspension, and Ji be the distance of the centre of gravity of a small quantity of mercury fx by the addition of which the pendulum is made to vibrate seconds exactly, to determine the approximate ratio of /i to m, L being the length of the seconds pendulum, and r the radius of the cyhnder con taining the mercury. The moment of inertia of the mercury n, which may be regarded approximately as a circular lamina of fluid, about any diameter, and therefore about a diameter parallel to the axis from which the pendulum is suspended will be { ixr', and therefore its moment of inertia about the axis of suspension will be Also the radius of gyration of the mercury m about a line through its centre of gravity parallel to the axis of suspension being k, the moment of inertia about the axis of suspension will be m (A* + F). Hence, by the formula for the Centre of Oscilla- tion, we have approximately {fxh! + 7nh) L = fx (A'" + \r^) + m (hr + F), But also we shall have hi = h^ + kr ; hence (jJi + >nh) L = fx (k'- + \r) + mhl, ^{h'{L-h')-\r]=h{l-L), m ^i Ah{l-L) m Ah' (L - h') - r^ 328 MOTION OF RIGID BODIES ABOUT FIXED AXES. (4) A bent lever, of which the arms are of lengths a and h, and the angle between them 0, makes small oscillations in its own plane about the angular point ; to find the length of the isochronous simple pendulum. JKequired length = ^^ (a* + ^a^"" cos Q + b*f (5) A uniform rod of given length is bent into the form of a cycloid, and oscillates about a horizontal line joining^^its extremi- ties ; to find the length of the isochronous pendulum. If a be the length of the rod, the length of the isochronous pendulum will be 1 a. (6) A pendulum consists of an indefinitely thin rigid rod OA, and a globe of which the centre is ^ ; to determine the point A', in the line OA, at which the centre of another globe must be fixed in order that the oscillations of the system of the two globes may be executed in the smallest time possible. Let OA = a, OA' = a ; also let r, r', be the radii, and m, m', the masses of the globes A, A'. Then a = — ■Imim + m) a +i m imr +mr')} . m \^ } m Euler ; Theoria Motus Corporum Solidonini,-p. 215. ( 329 ) CHAPTER VIII. MOTION OF RIGID BODIES. FKEE AXES. SMOOTH SURFACES. If a body be in motion about a Principal Axis/ and be acted on by forces which do not tend to perturb the direction of this axis ; then, the motion of the centre of gravity of the body remaining the same as if all the forces were impressed on the mass con- densed at this point, the Principal Axis will always remain parallel to itself as an axis of permanent rotation, and the angular acceleration about this axis will be the same as if it were a fixed axis. The discovery of the existence of three principal axes in every body as axes of permanent rotation is due to Professor Segner of Gottingen, by whom it was communicated to the world in a memoir entitled Specimen Tlieorice Turhinum, pub- lished at Halle in the year 1755. For the complete development of the theory of rotation about permanent axes, the student is referred to Euler's Theoria Motus Corporxim Solidorum, cap. viii., a work of the greatest value for those who wish to acquire pro- found views on the subject of the motion of rigid bodies. If a body be revolving at any instant of time about an axis which is not a principal one, this axis will not be one of perma- nent rotation ; the body will revolve successively about a series of instantaneous axes, the positions of which both in relation to the body and to absolute space are different. The solution of the great physical problem of the Precession of the Equinoxes, published by DAlembert" in the year 1749, unfolded a com- plete method for the investigation of the general problem of ' The fallowing is the definition of Princijjal Axes given by Eiilrr, Theoria Motus Corporum Solidorum, p. 175 : " Axes princi pales cujusque corporis Mint tres illi axes per ejus centrum inertise trariscuntes, quorum resprctn momenta iner'.iae sunt vcl maxima vel minima," ^ Rccherrhcs sur In Prvvessinn dcs Equinoxes, 17'J'J'. 330 MOTION OF RIGID BODIES. rotation. In the following year was published by Euler^ a memoir entitled Decouverte d'un nouteau principe cle Mechanique, the object of which was to investigate general formulae for the motion of a body under the most general circumstances of motion and force. The equations, however, expressing under the most simple form the general conditions of rotation, were first given by Euler* in the year 1758, who availed himself of the principles of simplification afforded by the recent discoveries of Segner'^ respecting the existence of the three Principal Axes of material bodies. The consideration of the genera! problem of rotation was resumed by D'Alembert, and presented under its most general aspect in the first volume of his Opuscules Mathematiques , pub- lished in 1761, where he expresses disapprobation of the title prefixed by Euler to his memoir of 1749, in consideration of his own investigations on the Precession of the Equinoxes. The sub- ject of rotation was thoroughly investigated and exemplified by Euler in his TJieoria Motus Corporum Solidoruni et Rigidorum, which appeared in the year 1767. The same subject was afterwards investigated by Lagrange* on more general principles of analysis. In the year 1777 appeared a memoir entitled 'A new Theory of the Rotatory Motion of Bodies affected by Forces disturbing such motion,^ by Landen," a celebrated English mathematician, in which he expresses himself dissatisfied with the conclusions of the great continental philosophers on the subject of rotation- The subject was again resumed by Landen,'' a few years after- wards, when he developes more fully his own views, and per- sists in his opposition to the doctrines of his predecessors. There is a memoir by Wildbore in the Philosophical Trans- actions for the year 1790, in which the subject is investigated under anew light : the conclusions of the author are unfa voui able to the cause of Landen, whose views are in fact now generally ' Memoires dt V Jcademie des Sciences de Berlin, 1750. * Ibid. 1758. ' Specimen Theurice Turbiuum, Mob. * Memoires de C Academic des Sciences de Berlin, 1773 ; Mi'canique /Ina- Itjtique, Seconde Panie, ection ix. * Philosophical Transaciions. 1777. « Ibid. 17K5. FREE AXES. SMOOTH SURFACES. 331 exploded. For further inforniation on the history of the theory of rotation and Landen's controversy, the student is referred to a memoir by Mr. Whe^y ell, in the second volume of the Cambridge Philosopliical Transactions, 1827. The investigation of Euler's general equations of rotatory motion has been effected with great elegance and simplicity by Mr. O'Brien, in the fifth chapter of his Mathematical Tracts, part i. Sect. 1. Single Body. (1) A rod PQ (fig 162) of uniform thickness and density, having been placed in a given position with one end upon a smooth horizontal plane OA, and the other leaning against a smooth vertical plane OB, descends in a vertical plane A OB by the action of gravity ; to determine where the rod will detach itself from the vertical plane. Let PG = a = GQ, G being the centre of gravity of the rod; let GH be vertical and equal to y at any time t of the motion ; OH - X, L QPO = (2); de ^ but y = a-c cos (p, and therefore, from (l), cf ■ cos ^ B - mg - mc df ,2 d'^d) . o . d' cos

.d(^ a — -—^ ^ C - k' -^ - 2^0 cos a sm ; \ dt J df ^ ^ but, initially, -j-= ^ and = jS ; hence there is = C - 2ag cos a sin /3, and therefore ,7.2 («* cos^ -f k^) -— = 2ag cos a (sin j3 - sin 0) . . . . (5), which gives the angular velocity of the beam for every position which it can assume during its descent. From the geometry it is evident that z = X - a cos = 2 ^^ sin a + a (cos j3 - cos 0), by (4), and 3/ = « sin

if therefore from (5) we could obtain in terms of t, we might determine the values of y and z at any time from the beginning of the motion. 338 MOTION OF RIGID BODIES. Again, for the pressure on the plane at any tune, we have, from (3), M = but, from (5), a cos df ' ddt sm 3 - sm d> -^ = 2ag cos a , ^ , df a cos- ^ + /;' 2 -f cos and therefore, differentiating with respect to t, and dividing by- cos ^, 1 f/^^ 2a'^ cos a sin ^ (sin /3 - sin -— x j dt df "^ dt dt df but from (6) and (7) it is clear that hence we obtain cos — = a ^ (-^) ' dt dt ^dx dx ^ . dx ^,^ d(t> d

f. d'^x . f. dx dO a -—■ = cos (/ -zrir - sm y -V- -r > df df dt dt d^x a d^ /, dx dB -T^ = 7. —r^ + tan B -^ -^, dt cos B df dt dt a d^(p a dB' , .^, 3:r. by(7); cos B df sin B df FREE AXES. SMOOTH SURFACES. 341 substituting this value of — - in (3), Ave obtain U' „\ c?V . a dB' + — cos t/ -rx = G' sm a + ^cos B a j df ^ sin B dt' ' and therefore, by (9), a^ 4 k^ cos^ B d''(t> . 2f/a' sin a sin^ B (I - sin 0) = /7 Sin n -4- * _j acosB df ^ cos' B (a' +J' cos' B) a- (1 + sin' B-2 sin' B) + k' cos" B ^ cos- (a' + k' cos' 0) hence, by (2), we have for the tension of the string for any posi- tion of the cylinder, „ mk' d'(p ,, . a" (\ + sin' 0-2 sin^ B) + W cos* B a df '' cos B {a' + ^' cos' B) Euler; Nova Acta Acad. Petrop. 1795; p. 64. (6) A uniform heavy rod OA, (fig. 167), which is at liberty to oscillate in a vertical plane about a horizontal axis through 0, falls from a horizontal position ; to determine the angle included between the direction of the rod and the direction of the pres- sure upon the fixed axis, for any position of the rod. From draw Oni at right angles to OA and to the fixed axis ; and produce A indefinitely to a point n. Let It, S, denote the resolved parts of the reaction of the fixed axis along Om, On, for any position of the rod. Draw Ox horizontal and at right angles to the fixed axis. Let OA = a ; ni = the mass of the rod ; Z. AOx = B, at any time t. Then, for the motion of the rod about its centre of gravity G, the moment of inertia about G being jr^ma', d^ df Y2 'tna' -j^ = 2 aR, d B ^ -rt , , ma -—= &R (1). dt Also, for the motion about O, the moment of inertia about O being i ma', (PQ 4 ma -—■ = mq . h a cos d, ^ df -^ ' 2a ~= 3^7 cos B (2). df 342 MOTION OF RIGID BODIES. Eliminating — - between (1) and (2), we get R = \mg co^ B (3). Again, equating S to the resolved part of the weight along OA and the centrifugal force, o • n f" fwdr d6~\ jS = mq sm a + r — - I '' Jo \ a (If J = mg sin + | ma -p; (4). dO Again, multiplying (2) by — , and integrating, do' ^ . ^ a -r-j = 6+3^ sm {) ; OjJj but -— = 0, when 0=0; hence C = 0, and therefore at dO"- „ . ^ a -— = 3(/ sm {J : dt- -^ hence, from (4), we get S = mg sin + i mg sin B = '% mg sin B . . . . (5). Let be the angle which the whole reaction of the fixed axis makes with the line 0)i ; then R tan = -5- • and therefore, by the equations (4) and (5), tan B tan = xg j which gives the value of for any position of the rod; (p is evidently the angle between the direction of the whole pressure on the fixed axis and the length OA of the rod. A solution of this problem was given in Chap. \\., by the direct application of D'Alembert's Principle. (7) A uniform rod, acted on by gravity, is oscillating in a vertical plane about one extremity ; to find the tendency of the vis inertia in any position to bend the rod at any point, and to ascertain the point at which this tendency is a maximum. Let OA (fig. 168) be the position of the rod at any time t', Ox an indefinite horizontal line through 0, the fixed extremity of the rod, in the vertical plane through OA. Take C any FREE AXES. SMOOTH SURFACES. 343 point in OA, P any point in CA. Let 0A = 2a, OC=c, OP = r, /.A0x = 9, k = the radius of gyration about ; fn = the mass of the rod. Then the fot'ce gained by an element dr of the rod at the point P, resolved at right angles to OP, will be equal to dr f d'O A m — r -— r - g cos U I : 2a\ df ^ / and the moment of this about C will be equal to ^(,.^_j,eose)(,--o)*-; hence the whole moment to produce bending at C will be equal to ^ p/ ^--j \ But, for the motion of the rod, we have d'd mJ^ -j-^ = mga cos 0, and therefore, 3 a~ being the value of k', d'O 3g a — V = -^ cos B. df 4a Hence the expression (1) becomes = m^^^ r {3 (,< _ c) - (4a - 3c)} (r - c) c/r = "^"^-^ {(2« - ^y - K4« - 3c) (2a - c)^} = '!l^{2a - of {2a- c -I (4a - 3c)} 8 a' e = '!^^^^c(2a-c)\ 16a' AVhen this expression is a maximum, we have (2a - cf - 2c (2a - c) = 0, 2a - 3c = 0, c = la, or 0C=10A. 344 MOTION OF RIGID BODIES. (8) An angular velocity having been impressed upon a hetero- geneous sphere, about an axis, perpendicular to the vertical plane which contains its centre of gravity G and its geometrical centre C, and passing through G (fig. 164), it is then placed upon a smooth horizontal plane ; to determine the magnitude of the impressed angular velocity that G may rise into a point in the vertical line SCK through C, and there rest ; the initial magnitude of the angle between CG and the vertical radius CS being given. Let CG = c, k = the radius of gyration about G, a= the initial value of the angle GCS, and w = the required angidar velocity; then w will be determined by the equation (A" -i- c^ sin" a) w^ = 2cg (l + cos a). Euler; Nova Acta Acad. Petr op. 1783; p. 119. (9) A chain, ten yards long, consisting of indefinitely small equal links, being laid straight on a perfectly smooth horizontal plane, except one part, a yard in length, which hangs down per- pendicularly below the plane ; in what time will the chain entirely quit the plane ? Time = 2.890663 seconds nearly. hady's and Gentleman'' s Diary, 1758, Diarian Reposi- tory, p. 683. Sect. 2. Several Bodies. (l) A wheel and axle is loaded with given weights P and Q, (fig. 169), which are not in equilibrium; to determine their motion and the tension of the strings by which the weights are suspended. Through C, the centre of the wheel and axle, draw the hori- zontal line A CB meeting the strings in A and B ; lei AC = a, BC = a' ; m = the mass of P, wj' = that of Q, pc = that of the wheel and axle together ; k = the radius of gyration of the wheel and axle about their common axis ; AP = x, BQ == x\ T= the tendon of AP, T = the tension of BQ ; 6 = the angle through which the wheel and axle have revolved at the end of the time t about their common axis. Then, for the motion of P, we have "'§ = "'^-^ ('>' FREE AXES. SMOOTH SURFACES. 345 for the motion of Q, , drx' , m, ^ X ^ -^=^9-T' (2); and, for the rotation of the wheel and axle, tj^k'^= Ta- To! (3). elf ^ ^ But, from the geometry, it is clear that dx ~dt~ hence, from (1) and (2), dz dS dx' , dB dt dt ' dt dt ma — = mg - T (4), - ^^a ~ = mg - T (5). Substituting the values of T and T from (4) and {b) in the equation (3), we get ;, d'o ( d'e\ , ,( ,d'e y-i^ —TT = nia \ q - a -—■ ] - ma [ g + a -—r df Y dif J Y df d'f) {tnd^ + m'a' + file) -— = g {ma - mo!) .... (6) ; whence B is immediately obtained in terms of t, the initial values of B and — - being supposed to be known. . From (4) and (6) we have rp _ mag {ma - m'a!) _ 7na' + m'a' + pt^" ' and from (5), (6), „, , m!a!q (ma - md) T = mg + — / , ,„ ji . ma + ma + (ji,k~ (2) Two equal uniform rods AC, BC, (fig. 170), having a compass joint at C, are laid in a line upon a horizontal plane- A string CDP having a given weight P at one end passes over a smooth pin D above the plane, and has its other end fastened to C which is vertically beneath the pin ; to determine the motion when P descends. 346 MOTION OF RIGID BODIES. Let AC = 2a = BC, L CAB =6; J? = the vertical reaction of the plane at each of the points A and B ; T = the tension of the string ; S = the mutual action of the two rods at the joint, which will evidently take place in a horizontal line parallel to AB; m = the mass of each of the rods, /x = the mass of the weight P. Let G he the centre of gravity of the rod A C; draw GH, CE, at right angles to AB ; let EH = x, GH = y, k = the radius of gjTration oi AC about G. Then, for the motion of the rod A C, we have, resolving forces horizontally, -'^"^ «' resolving vertically, \T being the force exerted by the string on each rod, ^7 2. . m"-^ = R + \T-mg (2); dr and, taking moments about G, m¥ - ^ = Sa sin + ^ Jf* cos B ~ Ra cos B. . . .(3). Also, for the motion of P, the increment of DP being double that of GH, ^-§-^^^ w- Multiplying the equation (2) by a cos B, we have ma cos B i-jz + g\ = {R + IT) a cos B, and therefore, adding this equation to the equation (3), mk^ -rr^ + ma cos B ( ~ + q] = Sa sin B + Ta cos B 1 df \df ^J hence, from (1) and (4), mk- -—7 + ma cos B —r., + ] = nia sm B — -^ + nacosBl a - 2 —^ ] ; df \ dtr "^y dtr \ df J and therefore, since x = a cos B, and y = a sin B, -,,d-B J „c?^sin0 . ^d/cosB\ „ , r^dSvuB mk -=^ 4 ma' cos t^ — — - sm t^ — =75— + 2jM,a cos y — r-^ — dt \ df dt ) dtr = ay {/x - m) cos B, frep: axes, smooth sukfaces. 347 f 1 7 2^ (^'0 n /I (r' sin B . ^ n {ma + mkr) -— + 2(xa- cos 9 j-^— = ag (fjL - m) cos (f. fid Multiplying both sides of this equation by 2 — , and integrating, we have, since -r = when = 0, dt (mar + mW + 2ju«^ cos" 0) — = lag (^u, - m) sin 0, which determines the angular velocity of the rods for any position. The value of — and therefore of -—, being known in terms of dt dt 0, we may readily obtain the values of M, S, and T, from the equations (1), (2), (4), in terms of the same angle. (3) A tube, moveable in a horizontal plane about a vertical axis, is charged with any number of balls at assigned intervals ; supposing a given angular velocity to be communicated to the tube, it is required to determine the motion of the tube and of the balls. Let a, a , a ,...be the initial distances of the balls from the fixed axis, and r, r',r,... their distances at any time t from the com- mencement of the motion. Let m, ni , ni' ,... be the masses of the balls, |U, of the tube ; and let Q be the angle through which the tube has revolved at the end of the time t. Let R, R' , R ,... denote the mutual actions and reactions of the balls and the tube. Then, ptZ;" denoting the moment of inertia of the tube about the vertical axis, we shall have, for the motion of the tube, d^ df Also, for the motion of the balls, m, m', m ,... we have «5=ijy, »^^ = -iJ- (2), df' r dt r m'—=R'^-, m ^=-R -, (3), dt r dr r „d^oo' -r,„y" „d?y -D.,x . . ^' —-.&• + Sr' +K'r" + ■.(!)■ 348 MOTION OF RIGID BODIES. where {x, y), (x, y'), {z, y),..- are the rectangular co-ordinates of the balls at the time t. Multiplying the former and the latter of the equations (2) by y and x respectively, and subtracting the latter from the former of the resulting equations, we get / d'x d\\ „ y df dfj and therefore, since x = r cos $, y = r sin B, the axis of x being supposed to coincide with the initial position of the tube, we may readily obtain, by substitution, dt\ dt ) In like manner, from the equations of (3), (4),. ..we may get '""dti'd^) = -'''' : d( ,de\ „, „ Hence from (1) we have j,d'e d( .do ,,.dd , „.de dt dt\ dt dt dt integrating, we get 12 f^^ ^ , 2 ' /•> ' o X dQ iKiV' -— = C - \inr -\- m7'' + mv + •■•) — ; dt ^ ' dt cW dt JO but, supposmg w to be the initial value of — , we have (xk-b) = C - {md^ + md^' + nid' + .•■) w ; , d% nk^ + md^ + m'a'^ + md'^ + . . . hence -31=^^ 2 rr^ tt^o w . . . . 5). dt fifc' + mr + mv + mr + .. Again, from the equations (2), we have cPx drxi and thence, substituting for x and y their values in 7- and Q, d-r dQ' lie='df ^'^- FREE AXES. SMOOTH SURFACES. 349 In the same way, from (3), we may get df^ df' dfi and therefore, eliminating — between these two equations, ,d^_ dV ^ df ~ '' d€- ' integrating and bearing in mind that both — and — are initially equal to zero. , dr dr r -— = r -zr , dt dt and therefore — = — ; r r' integrating again we have, a, a , being the initial values of r, r', r r , a - =—, ) r = — r. a a a In precisely the same way it may be shown that a ,„ a' r = — r, r = — r, . . . a a Hence from (5) we have f/0 yjk^ -^ ma^ + ma' + md'^ + etc 1 ^ f 9 t l'> " '■"> \ ' (j.k' + ( ma + ma + ma + ... ] ~ V /« From (6) and (7) we obtain (fr _ 2 (f^^^ + ^«" + m'a'^ + m'a~ + etc \ 1.2 I 2 / /"> " "2 I ' fxK + ( 7?ia + ma +ma + (D- Multiplying both sides of this equation by 2 — , integrating, dv and bearing in mind that -j: = ^ when r = a, we shall easily see that _ = a>^(r^-«^)il_ ^ p.... (8). u^k- + Cma' + w«'^ + m V^ + ...) — a^ 350 MOTION OF RIGID BODIES. The equations (7) and (8) will give us, for any assigned distance of the ball m from the axis of rotation, the angular velocity of the tube and the velocity of the ball m within it. If between (7) and (8) we eliminate dt, we shall obtain the differential equation in polar co-ordinates to the path of rti in the horizontal plane passing through the axis of the tube. Similar results may evidently be obtained for the other balls with w^hich the tube is charged. Cor. If /x = Oj the equations (7) and (8) become dQ a"w dr^ , , ,, a'w^ and therefore, eliminating dt, 55 = (r - a') -3 , dO d": r dS^ a^ f o 2,5 / d r (;•" - ay ['^ ~ ~~ integrating, and remembering that 6=0 when r = a, o = cos - , - = cos 6. r r Again, to determine the relation between r and t, Ave have J. \ rdr , 1 , , ' i dt= — , t = — (r - ay, v~ and therefore — = i + urf. a Similar relations holding good for the other balls, Ave have, for the equations to their paths, a a a ^ -=-,=—, = .... = cos u, r r r which shew that they all move in straight lines at right angles to the initial position of the tube ; and for their distances from the axis of rotation at any time, r' r' r' „ „ - = — = -TT= = 1 -y^t\ a a' a FREE AXES. SMOOTH SURFACES. 351 Clairaut; Mem. cle VAcad. des Sciences de Paris, 1742, p. 48. Daniel Bernoulli; Mem. de VAcad. des Sciences de Berlin, 1745, p. 54. Euler; Opusctda, de motu corporum tuhis mohilibus inclusorum, p. 71. (4) A heavy particle P descends down a smooth inclined plane BA, (fig. 171), forming the upper surface of a solid BAC, which is capable of sliding freely along a smooth hori- zontal plane OAx ; to determine the motion of the particle and of the body, both of which are supposed to have initially no motion. ■ Let PM be at right angles to Ox, and let B be the point in the inclined plane which the particle occupies initially ; let A be supposed to coincide with at the commencement of the motion. Let OM = x, PM = y, OA = s, AB = a, BP = s , L BA C = a; and R = the action and reaction of the plane and the particle. Then, m denoting the mass of the particle and rn of the body, we shall have m --^ = li cos a - mq ( 1 ), dr m — = - P smci (2), ,cPs. m -— - = M sm a (3). df But y = {a ~ s) sin a, x = s + {a - s) cos a ; hence, from (1), and, from (2), m Adding together (3) and (5), we have (m + m) -j^ - m cos a -^pr = . . . . (6). dt clr Multiplying (3) by cos a and (4) by sin a, we have, adding together the resulting equations, , el's . ^ a s . ,^v m cos a -— + m sm a — = tng sm a . . , . (7;. (ps' ni sin a -—J = my - R cos a (4), Ct S CI S -y-v • / ^ \ m — - - m cos a -—^ = - li sm a . . . . (5). dir dt 352 MOTION OF RIGID BODIES. Multiplying (6) by sin^ a, (7) by cos u, and adding together the resulting equations, {m sin^ a + m!) —pr = nig sin a cos a ; integrating twice with respect to t, and bearing in mind that ds 5 = and -y- = when ^ = 0, we obtain , ,„ m sin a cos a s = l9^ --2 -, (^)- m sin a + m Again, multiplying (7) by m + m! , (6) by ni cos a, and sub- tracting the latter of the resulting equations from the former, we have m {m sin" a + 971!) -j-j = ^nff sin a (m + ni'), d~s' g sin a (m + m) _ f/^* m sin" a + m' integrating twice, and recollectinoj that s -0 and — r- = Avhen i = 0, we get ^,^^ .(m+^>in_°.... (9). m sm" a + m The equation (8) gives the position of the moveable inclined plane, and (9) the place of the particle on the plane at any time. Again, by (3), P m' d~s mnig cos a sin a df m sin^ a + m" which gives the value of the mutual pressure of the particle and the plane ; the value of which, therefore, is invariable. John Bernoulli; Comment. Acad. Petrop. 1730, p. 11. Ojiera, tom. iii. p. 365. Euler; Opuscula, de motu corporum tuhis mohilibus inclusorum, p. 28. (5) A heavy particle is placed within a thin tube APB, (fig. 172), situated in a vertical plane, which passes through a horizontal line OE; the tube is attached rigidly to a body ABC, the lower surface of which is flat, and in contact with a smooth horizontal plane, along which it is able to slide freely ; FREE AXES. SMOOTH SURFACES. 353 supposing the particle and the body to be initially at rest, to find their subsequent motions. Let A be the point of the tube which the particle occupies initially, and O the initial position of the point B of the body ; let OB = s, length AP of the tube = s' ; 0M= x, P3I= y, where PM is vertical ; ^ the inclination to the horizon of an element of the tube at P ; m = the mass of the particle and ni = the mass of the body ; B, = the action and reaction of the tube and the particle. Then, for the motion of the particle, we have m — ^ - Esintp (1), m -yy = i2 COS (p - mg (2) ; and, for the motion of the body, m -— = Mm\\(b (3). Again, from the geometry it is evident that dx = ds - cos (^ ds (4J, and dy = - sin

, Civ itV A A 354 MOTION OF RIGID BODIES. and therefore, by (5), (Ps . d ( . ds'\ m cos (p -y^ + m sm — I sm ^ — ) = mg sin ; 1 ^ r /^N -— = - c<7 sin ^ ; multiplying by 2 -^ , and integrating, (a^ + ^^ + c' - 2«c cos being by the hypothesis small angles, we may neglect their second and higher powers in the equations, and we get ^^ m— = mg-R-F9, m^ = -R9 + F, dr nik' ^ = -Rc(9+(l>) + F(c- h). Eliminating R and F between these three equations, we shall finally obtain, as far as the first order of small quantities, *'^-.(«^0) = (§ + .e)(-*) a> But, from the geometry, it is clear that y = {a -h)s\xi 9 - c sin (l> ={a -h)9 - C(^ nearly, and therefore, putting for brevity a - h =^ e, d^_ cP9_ d^^ df '"'df "" df' FREE AXES. ROUGH SURFACES. 365 hence the equation (1) becomes ik' + c^~cb)^-e(c-b)'^+cg-,bge = 0... .(2). Now, since there is no sliding, we may shew, by precisely the same method as in the case of problem (2), that

t -r^ = q sm wr -— ; df df ^ a df but, from (4), we get ^'df^'df' , ^ d'^x . ^ d~y . , F c?V ,. hence cos tot -r-^ + sm wt -~ = q sm wt — „ ^r^ (6). df df ^ d df ^ ' From the geometry it is clear that X = r cos wt + a sin wt, y = r sin wt - a cos wt ; differentiating these expressions twice with respect to t, we shall get —-; = cos wt -r-^ - 2w sin wt — — wr cos wt - aw' sin wt, dt df dt ^y • J. ^^»' ^ .dr . . , -^ = sm wP -— 5 + 2w cos wt - — wY sm wt + aw~ cos wt : df df dt FREE AXES. ROUGH SIRFACES. 369 substituting these expressions for — -^ , — ^ , in the equations (5) and (6), we obtain 2w — - + r/w = a cos w^ (7), at m , a' + k' (Pr 2 • , .„^ and — — ^ - o) r = g s\n wt (8). a at' Since d^ = 2k', the equation (8) becomes I* / o 2 o • J df ^ '^ the integral of this equation is r = -% sin wf + Ce*"" + C f--", where w' = (f/w, and C, C, are arbitrary constants. If we dr determine C and C from the conditions that r = 0, — - = 0, initially, we shall have which determines the position of the cylinder at any time before it detaches itself from the revolving plane : differentiating with respect to t, r = - ^, sin iot + -^^- (£-" - £-^'0, 5w i * = _?2 cos »«.f (.-+<-"■•); hence, from (7), we obtain ~= a cos lot - aw +%q cos w^ ^ (s'"" + e"""'') = §<7 cos w^ - ««' - ^ (£"'" + E-""0, o which gives the value of R for any time of the motion of the cylinder upon the plane : when R= 0, or when the cylinder leaves the plane, 9<7 cos wt = oaJ" ■+ 2g (s'"'' + s''"'% an equation which fixes the epoch of the separation. BB 370 MOTION or RIGID BODIES. (7) A sphere is projected directly down an inclined plane with a motion both of translation and of rotation ; the motion of rotation is the same in point of dii'ection as that which would correspond to perfect downward rolling, but greater in magni- tude : to determine the motion of the sphere, having given the coefficients both of statical and of dynamical friction between the sphere and the inclined plane. Let OA (fig. 182) be the incliiaed plane, Cthe position of the sphere's centre, and M its point of contact with OA at the end of a time t from the beginning of the motion. Let fj. = the coefficient of dynamical friction between the sphere and the plane, a = the radius of the sphere, OM = s,

t, cos a) + ct, uaqf

a df d + k^ which is the value of the statical friction necessary to secure perfect rolling in the second stage of the motion. If the statical friction be less than this, dynamical friction wiU arise, and will evidently exert itself jip the plane. Hence, for the motion, ^^ -^ = g (sin a - /x cos a) (A), J^ -^ = fxag cos ct (B). It may be easily ascertained that the coefficient of g in the drs . . . expression for -^ is positive ; for the coefficient of friction FREE AXES. ROUGH SURFACES. 373 necessary for perfect rolling F ^ F _ k^ tan a R mg cos a a^ + k^ ' andj since />t is less than this by hypothesis, we have (x L —^ -^ tan a, and therefore y. L tan a, /«. cos a L sin a. From (A) and (B) we have ds , . d(p y^agt ds , . , ~ = gt (sm a - fi cos a) + c , dt F cos a + w . It may be readily seen that — — a -^ never becomes zero in ^ ^ dt dt the second stage of the motion, but is always positive ; for, bear- ing in mind that c' = aw', ds d(h ,f . a^ \ a^ + k^ f k"^ \ = ^^cosa-^^(^-^-^tana-^J; hence the sphere will always rotate too slowly, in comparison with the velocity of translation, to correspond to perfect rolling. If 5' denote the space tlirough which the sphere descends along the plane in the second stage of the motion, in conse- quence of sHding, ds ds d i fiag cos a the centre of the sphere will remain stationary, and that at the end of this time the angular velocity of the sj)here will become zero. Before proceeding to investigate the nature of the motion after the end of the stationary period of the centre of the sphere, it will be necessary to determine the amount of upward friction requisite to cause the sphere subsequently to assume a FREE AXES. ROUGH SURFACES. 375 motion of perfect rolling. The coefficient of friction requisite for this purpose, (see the preceding problem,) is equal to W tan a But /M, is equal to tan a, and therefore exceeds the requisite magnitude. The sphere will then proceed subsequently to roll down the plane without sliding ; and the space described by its centre, at the end of a time t from the termination of its sta- tionary interval, will be equal to 2 c^gf sin a which, since k^ = la^, is equal to ^-^gf sin a. Also, putting for k^ its value, the stationary interval will have for its value — ; — . bg sm o Euler; Acta Acad. Petrop. P. ii. p. 131 ; 1781. (9) A homogeneous sphere attracted towards a given centre of force varying directly as the distance, is projected with a given velocity along a plane passing through that centre, friction being such as to j^revent all sliding ; to determine the path described by the sphere. Let 0, (fig. 1 83), the centre of force, be taken as the origin of co-ordinates, and let Ox, Oy, which are at right angles to each other and in the plane along which the sphere rolls, be taken as the co-ordinate axes. Let C be the centre of the sphere at any time of its motion, P its point of contact with the plane xOy; join CO,CP, and draw PiUf parallel togO ; draw also Oz at right angles to the plane xOg. Let 0C= r, CP = c, OM = x, PM -- y,m== the mass of the sphere, ^a. = the attraction of the central force u2:)on a unit of mass collected at a point at a unit of distance ; let X, Y, denote the friction of the plane on the sphere, estimated parallel to Ox, Oy ; J, w', the angular velocities of the sphere at any time about diameters parallel to Ox, Oy, estimated in the direc- tions indicated by the arrows in the planes yOz, zOx; k = the radius of gyration of the sphere about a diameter. It may be readily ascertained that the attraction on the whole sphere will be equal to a force fimr in the direction CO, and the 376 MOTION OF RIGID BODIES. resolved parts of this force parallel to Ox, Oy, are evidently - ixnix, - fxmy. HencOj for the motion of the sphere, we have m —-J = X - fA.mx (1), ^'cff ^^^^ ^^' mk' ^ = cY (3), at mk'% =-cX (4).* dt ^ ^ But, since the friction is sufficiently great to prevent all sliding, it is clear that dx „ dy hence, from (3) and (4), we get mk^ d'^y -^ _ mJ^ d^x * If to', to", o)'", denote the angular velocities of a rigid body about three straight lines through its centre of inertia, parallel to the axes of x, y, z, respectively, and ^m denote an element of the mass, then, (See Pratt"s Mechanical Philosophy, First Edit. p. 428), {J"' - u,'") S (dfn.yz) + (u,"'uj' - ^) S i^m.xy) - fioio' + "tJ) s {Sm.xz) + a;"V S(gmy) - w'io'" S (Sm.z') + -^' 2 {Sm . (/ + z')} = L, where L denotes the moment of the forces about the first of the three straight lines. Similar equations, mutatis mutandis, will hold in relation to the axes of y and z. if the body be a homogeneous sphere, as in the present j)roblem, S(8m.y;2)=0, S(gm.^y)=0, S(Sm..r^)=0, ^(Sm.y^)-'2iBm.z^)=0, and therefore, adopting the notation of the text. mkr —- = L, dt L being equal to cY. Similarly mU' ■—- = - cX. dt FREE AXES. ROUGH SURFACES. 377 and therefore, from (1), (2), c" + k^ cVx & elf '^' c^ + W d-y t or, since k^ = ?c^, -de^-'-y'' integrating, and adding arbitrary constants, dx" 5/x dy- 5^ df 7 ' df 7 ^ ' w.?^ dij let a, J, be the initial values of x, y, and a, j3, of — - , -~; then dt dt 7 7 and therefore eliminating dt, and putting dy^ dx^ dy dx o = — + a , = -^ + 6 , 5fl OfX we sret *"~^' a" :y'' (i''-2/')' K-^T integrating, we have sin"^ -. , i where c is an arbitrary constant ; hence sin"' Y, = sin'' — + cos'' c (5), o a y ex ,. ,^\l , 0? a \ a J c'V ^ , X y y'^ ,, ,2N /, x'\ -,3 - 2c' -, I + 15 = (I - O 1 - - a abb \ a J — , - 2c' - f, + f- - 1 - c'- : o78 MOTION OF RIGID BODIES. putting a, h, for x, y, we have for the determination of c, — - 2c - - + — = 1 - c ' ; a abb let m, n, be the two values of c' given by this quadratic; then for the equation to the path of the sphere we shall have o^ a b b^ or 2^ - I + ^ = 1 - ^^^ a a b b Thus, m and n being neither of them greater than unity, we see that the sphere will describe one or other of two ellipses, the centres of which coincide with the origin of co-ordinates. The ellipse, which is to be taken, will depend upon the sign of the ratio a to |3. Sect. 2. Several Bodies. (1) A cylinder rolls down a perfectly rough inclined plane, while a string coils round it which unwinds from an equal cylinder revolving about its axis which is fixed, the position of the latter cylinder being such that the string is parallel to the plane; to find the accelerative force of descent, the tension of the string, and the friction of the inclined plane. Let (fig. 184) be the centre of gravity of the descending cylinder at any time of its motion down the plane BA, 31 being its point of contact with the plane ; let C be the centre of gravity of the other cylinder ; join CO. Let CO = x at any time t ; a = the radius of each of the cylinders, a = the incKnation of the plane BA to the horizon, T= the tension of the uncoiled string, F = the friction of the inclined plane exerted upon the cylinder at Ji" in the direction 3IB ; m = the mass of each of the cylinders, and k = the radius of gyration of each about its axis ; let B, 6', denote the angles through which the cylinders O, C, have revolved about their axes at the end of the time t. FREE AXES. ROUGH SURFACES. 379 Then, for the motion of the cylinder O, we have m— = mg sin a- F-T (1), mk'^ = (F-T)a (2); and, for the motion of the cylinder C, mk'^= Ta (3). de ^ ' Multiplying the equations (1) and (3) by a and 2 respectively, and adding the resulting equations to (2), we get ,2 ; and therefore, by (3), -m]^^-^=Ta'-Fa' (4); dt- ^ ^ also, from (2), -ma^'^^= Ta' + Fa' (5), and therefore, adding together these two last equations. -m{a' + k')^=2Ta' (6); and therefore, by (1), 2m'a' -^- m{a' + ¥) -^ = 2m'aV/ : FREE AXES. ROUGH SURFACES. 381 but, if / denote the original length of the free string, it is clear that , d'^x d\i d'(t> d\ d\i ^ d'^x hence we have {4m'a^ + w (a' + k')] Va" = - 2wi'a^^ .... (7) ; integrating, and bearing in mind that — = when ^ = 0, dt integrating again, and taking c for the initial value of x, m'a^gf 4mar + m(ar + kr) We may easily get also 2m'a'gf y = I - c + and therefore, since a

n ->r m) —-^ (7). df dr Again, from (1) and (2), m cos a -r^ + m san a -~ = r - mq sm n, df df ^ and therefore, by (3), d^x . dry ,, d^B . • a cos a -TTT + a sin a — = A" -r-;r - «<7 sm a : substituting in this equation the expressions for -yf and -^ , df df dt is equation given in (6) and (7), we obtain ijfx {m'd^ cos^ a + {771 + m!) (a" sin' a + k"^)} — - = - m'a^g sin a cos a : which gives the value of —- , which it appears therefore is con- stant during the whole motion ; the values of -—■ , —f , -— • , df df dt may now be readily obtained by the aid of the equations (5), (6), (7), and will be constant during the whole motion. XT • ^1 1 n dx dry dO ^x . ,. jvnowing the values ot -^ > ~f^ > -rs y -^ j ^^^ may immedi- ately obtain the values of x, y, 6, x', in terms of t, if we have - ... dx dx given the initial values of x, x' , — , — . The values of R and ^ ' dt dt Fma.y also be readily obtained from the equations (]),(2), (3), (4). (4) A bullet is fired with a given velocity into a body in a direction passing through the centre of gravity of the body ; the body is initially at rest and is capable of free motion, not being under the action of any forces ; to determine the velocities of the bullet and of the body when the bullet has traversed any space within the body ; the resistance of the body to the motion of the bullet being supposed to be a constant force. Let k denote the constant retarding force, m the mass of the bullet, fM of the body, /3 the initial velocity of the bullet ; then if FREE AXES. ROUGH SFRFACES. 385 ti and V denote the velocities of the bullet and of the body when the bullet has traversed a space x within the body, « - — '- — -L —^ — i3 (7n + i^)x\ , Camus ; Mem. deVAcad. des Sciences de Paris, 1738, p. 147. c ( 386 ) CHAPTER X. DYNAMICAL PRINCIPLES. Sect. 1. Vis Viva. The term Vis Viva was first introduced into the language of Mechanics by Leibnitz, in a memoir published in the Acta Eruditorum for the year 1695, entitled Specimen dynamicum pro admirandis 7iaturcB legibus circa coiporum vires et mutuas actiones detegendis et ad suas caiisas revocandis : it was intended by its author to signify the force of a body in actual motion, called otherwise its Vis Motrix or Moving Force, as distin- guished from the statical pressure of a body, which has merely a tendencj^ to motion, against a fixed obstacle ; the statical force of a body he designated by the appellation of Vis Mortua. Leibnitz contended, in opposition to the received doctrine of the Cartesians, that the proper measure of the Vis Viva or Mo^^ng Force of a body, is the product of its mass into the square of its velocity, the measure adopted by the disciples of Descartes having been the same as that of the Quantity of Motion, namely, the product of the mass and the first power of the velocity. This contrariety of opinion in respect to the estimation of Moving Force, gave rise to one of the most memorable controversies in the annals of philosophy; almost all the mathematicians of Europe ultimately arranging them- selves as partizans, either of the Cartesian or of the Leibnitzian doctrine. Among the adherents of Leibnitz may be mentioned John and Daniel Bernoulli, Poleni, Wolflf, 's Gravesande, Camus, Muschenbroek, Papin, Hermann, Bulfinger, Kcenig, and even- tually Madame du Chatelet ; while in the opposite ranks may be named Maclaurin, Clarke, Stirling, Desaguliers, Catalan, Robins, Mairan, and Voltaire. The Vis Motrix, or, as Leibnitz expressed it, the Vis Viva of a moving body was regarded as DYNAMICAL PRINCIPLES oO / a power inherent in the body, by which it is able to encounter a certain amount of resistance before losing the whole of its velocity: the question reduced itself, therefore, to the deter- mination of an appropriate measure of this amount of resistance, to which the Moving Force was supposed to be proportional. Leibnitz regarded the product of the mass of the body and the space through which it must move, under the action of a given retarding force, to lose the whole of its velocity, as the correct measure of the whole resistance expended in the destruction of its motion, and therefore as a proper representative of the Vis Motrix or Vis Viva of the body. Now, by the theory of uniform acceleration, mio" = 2mfs, m being the mass of the body, and s the space which it must describe, under the action of a constant retarding force /, to lose the whole of its velocity v : hence it is evident that, according to the doctrine of Leibnitz, 77iv'^ will represent the body's Vis Viva. On the other hand, the Cartesians estimated the whole resistance necessary for the destruction of the body's velocity by the product of the mass of the body and the whole time of the action of the given retarding force ; and there- fore, by the formula mv = mj^i, it would follow that mv is the proper measure of the Vis Motrix, or in the language of Leibnitz, of the Vis Viva of the body. The memorable controversy of the Vis Viva, after raging for the space of about thirty years, was finally set to rest by the luminous observations of D'Alembert in the preface to his Dyiiamique, who declared the whole dispute to be a mere question of terms, and as having no possible connection with the fundamental principles of Mechanics. Since the publication of D'Alembert's work, the term Vis Viva has been used to signify merely the algebraical product of the mass of a moving body and the square of its velocity, while the words Moving Force have been universally employed, agreeably to the definition given by Newton in the Principia, in the signification of the product of the mass of a body and the accelerating force to which it is conceived to be subject, no physical theory what- ever in regard to the absolute nature of force being supposed to be involved in these definitions. For additional information respecting the controversy of the Vis Viva, the reader is referred to Montucla's Histoire cles Maihematiqucs, torn. iii. ; Hutton's cc2 388 DYNAMICAL PRIKCIPLES. Mathematical Dictionary under the word Force ; and Whewell's History of the Inductive Sciences. The Prmciple of the Conservation of Vis Viva is compre- hended in the following proposition : If a system of 2ici,rticles, any numher of which are rigidly connected together, mote from one position to another, either tvith or without constraint, under the action of finite accelerating for^^rs, external or internal ; the change of the vis viva of the whole system will he independent of the actions of the particles arising from their mutual connections, and will he equal to the sum of the changes tvhich wotdd he expe- rienced by the vis viva of each particle, were it constrained to move unconnectedly from its original to its new positio7i, under the action of the very accelerating forces to which it is subject in the actual state of the motion. This Principle immediately furnishes us with a first integral of the differential equations of motion, which is frequently of great use ; especially if the co-ordinates of the position of the moving system involve only one indepen- dent variable, as in the problem of the Centre of Oscillation, when the Principle is sufficient for the complete determination of the motion. The Principle employed by Huyghens' as the basis of his investigations on the problem of the Centre of Oscillation, con- stitutes under an indirect form a particular instance of the Principle of the Conservation of Vis Viva. John Bernoulli', however, was the first who enunciated the theory of the Conser- vation of Vis Viva, a name which he gave to the Principle, as a general law of nature, from which he deduced that of Huyghens as a particidar case. Daniel Bernoulli^ afterwards extended the application of the Principle to the motion of bodies subject to mutual attraction, or solicited towards fixed centres by forces varying as any functions of the distances. A demonstration of ' Si pendulum 6 pluribus poiideribus conipositura, atque e quiete dimissum, partem quamcunque oscillationis integrae confecerit, atque inde porro intelligan- tur pondera ejus singula, relicto communi vinculo, celeritates acquisitas sursum convertere, ac quousque possunt asccndere ; hoc facto, centrum gravilatis ex omnibus compositr-B, ad eandem altitudinem revprsum erit, quam ante inceptam oscillationem obtinebat. Horolog. Oscillator, p. 126. * Opera, passim. • ^ Memoires de VAcadhnie des Sciences de Berlin, 1748. DYNAMICAL PRINCIPLES. 389 the Principle in particular cases was first given by D'Alembert' by the aid of his general Principle of Dynamics, the same method of proof being, it was evident, of general application. (IJ A uniform rod AB (fig. 187) moves in a vertical plane, within a hemisphere ; to determine its angular velocity in any of its positions, its initial position being one of instantaneous rest. Let be the centre of the sphere ; G the middle point of AB, which will be its centre of gravity ; GH a perpendicular from G upon the horizontal radius through 0, which is in the plane of the rod's motion; let OG = c, AG = BG = a, k= the radius of gyration about G ; 011= x, GII=y, and = the angle of inclination of AB to the horizon at any time t. Then, by the Principle of the Conservation of Vis Viva, m being the mass of the rod, de di- df-j ^^ let h be the initial value of ?/ : then, since -^ ,-t- , t ^ ai'e initi- -^ dt dt dt ally zero, we have = C + Irayh ; , dci? dir ,o dQi"' , ,. hence -r^ + -fr + l^" — ^ = 2<7 ( w - h). dt' dt at "^ "^ But from the geometry it is plain that X = c sin 0, y = c cos 0, , dx r, dd dii . f. dO whence —r = c cos u -r , -f = - c sin a --- ; dt dt dt dt we have, therefore, 7/32 (c^ + k') — - = 2cg (cos - cos a), a being the initial value of 6; hence, putting for k' its value la', we have, for the angular velocity of the rod in any of its positions, ^^, icg , . (2) A rod PQ (fig. 188) is kept in a vertical position by means of two small rings .1 and A'; its lower end P is sup- ported on an inclined plane BC, which is at liberty to move ' I'laite de Dyiiainique, Seconde Partie, chap iv. p. 252. 390 DYNAMICAL PRINCIPLES. freely on a horizontal plane ; to determine the motion of the rod and the plane. Produce QP to meet the horizontal plane in the point ; let OP = y, OB = X, at any time of the motion ; h = the initial value of y, a = the inclination of the inclined plane to the vertical, m = the mass of the rod, wi' = the mass of the inclined plane. Then, by the Prmciple of the Conservation of Vis Viva, but, supposing the rod and the plane to be initially in a state of instantaneous rest, o = C - Iqmh • dy? dv^ hence rd -— -\- m -^ = 2ym (/* - y) ; (Xv Civ but, from the geometry, dx dii hence we have X = y tan a, -^ = tan a -p , at dt (ni tan" a + ni) -~ = 2mg (h - y), - dv - {m! tan* a + mj "^ — = - (2myJ dt, (h-yf the negative sign being taken, because y decreases as t increases : therefore, by integration, 2 (ni tan' a + tnf (h - tjf = C + {2mgy t ; but y = h when ^ = ; and therefore C = ; hence 2 (?n' tan* a + m) (Jt - y) = nigf, and therefore, for the value of y at any instant of the motion, , 5 mqf V = h ^— f — ^— ; m + m tan a and therefore, for the value of x, J I mqf tan a X = h tan a — ^" — -~ ^- . m + m tan a (3) AB (fig. 189) is a uniform beam, capable of moving freely about a hinge A ; the extremity B rests upon an inclined DYNAMICAL PRINCIPLES. 391 plane CE, which forms the upper surface of a body ECD ; the body rests with a flat base upon a smooth horizontal plane passing through A, the vertical j)lane which contains AB being supposed to cut the plane surface of the body CED at right angles, and to pass through its centre of gravity ; to determine the motion of the beam and the body. Let G be the centre of gravity of AB ; draw GH at right angles to the straight line A CD ; let m, m , denote the masses of the beam and of the body; AH=x, GH=y, L BAC= d, L ECD = a, AC= X , k==- the radius of gyration of AB about G. Then, by the Principle of the Conservation of Vis Viva, (dx^ dii^\ ,, dQ' , dx'^ ^, le* ie) '■"'''' df-""' If- '^'^"^y-' but from the geometry we see that X = a cos B, y = a sin B, x = -; sin (a - 0). . . . ( 1 ) ; sm a hence we have / o 7-,. dB 4:a , , , f.. dB y^ ^ • ^ m ia' + k') -— ; + . , - m cos" (a - t/) ^-, = C - 2mqa sm B, ^ ^ df sm-a ^ ^ df -^ dB' {m (a^ + k^) sin^ a + Ama^ cos^ (a - B)} -yr = sin" a{C - Imcja sin B) ; dB let /3 be the value of B w hen -— = ; then dt = sin" a{C - 2mga sin j3), and therefore we get JCR {m (a-+^-) sin" a + 4m'«^ cos^ («- ^)} -r? = 2mag sin^ a (sin j3 -sin B), dB' which gives the value of — for any assigned value of 0; whence, by the aid of the equations (1), we may obtain the values of dx dy dx ^ . . n ■, ^ -T- } ~r } -7-3 lor any position 01 the beam. dt dt df (4) A uniform lever ACB, (fig. 190), of which the arms AC and BC&re at right angles to each other, rests in equilibrium when AC is inclined at a given angle to the horizon ; if ^ C be raised to a horizontal position, C being fixed, to find the angle through which it will fall. 392 DYNAMICAL PRINCIPLES. Let CA = 2a, CB = 2a' , m = the mass of AC, m = the mass oi BC ; let B, & , be the inclinations of CA, CB, to the horizon, at any time of the motion. Then the \-is viva of the lever will be equal to 2mag sin B + 2m!a!g sin & + C; but, when B = and therefore 0' = 5 tt, the vis viva is equal to zero ; hence q = 2r)^a'g + C; hence the \ds viva for any position of the lever is equal to 2mag sin + 2m' a g sin B' - 2m' a g. Now, when the value of is a maximum, the vis viva will again become zero ; hence, for the required value of B, ma sin B + m'a' sin 0' = m'a' (1). Let /3, j3', be the values of B, B', for the equilibrium of the lever ; then ^^^ ^Qg ^3 ^ j^'^' ^^g ^' . hence from (1) there is cos /3' sin B + cos j3 sin B' = cos /3, or, since /3' = 5 tt - j3, B' = hw - B, sin /3 sin B + cos /3 cos = cos j3, cos (B - (5) = cos )3 ; and therefore B = 2j3, the angle thi'ough which CA falls. (5) To determine the motion of a pendulum, the axis of which is a cylinder resting upon two perfectly rough planes which coin- cide with the same horizontal plane, the cylindrical axis being thus capable of rolling along the planes. Let C (fig. 191) be the centre of a vertical section of the cylindrical axis made by a plane containing the centre of gravity of the j)endulum ; C may be regarded as the centre of gravity of the axis. Let G be the centre of gravity of the pendulum and cylinder together, and t7ik^ their moment of inertia about a horizontal line through G parallel to the axis, 771 denoting the sum of their masses. Let GH be drawn at right angles to the horizontal plane along which the axis rolls ; let be the point of contact of the section C of the axis M'ith this plane at any time of the motion, A being the position of corresponding to the equilibrium of the system. Let CO = c, CG = a, L G CK = ^, C7f being a vertical line, AH = x, GII= y. DYNAMICAL PRINCIPLES. 393 Then the vis viva of the system at the time t due to the mo- tion of G will be m f — ^ + -^ | , and the vis viva due to rotation [df ^ cie J about G will be mk~ -~ ; hence the whole vis -vdva of the system df ^ will be equal to (,, d(p^ dx^ dy'\ df df df ) also the sum of the prodvicts of the mass of each molecule of the system into the vertical space through which it has descended, will be equal to my together with some constant quantity de- pending upon the initial circumstances of the system. Hence, by the Principle of Conservation of \"is Viva, /,o d<^ dx' dfr\ ^ \ df df df J -^ ^ but fi'om the geometry it is evident that X = a sin ^ - c^, y = « cos - c, and therefore dx , ^ d(p dii . d hence we have (a^ -^ (? + W - 2ac cos 0) -^ = C" + 2(1 {a cos 9 - c); df let a be the maximum value of ^, then Q = C + 2g (a cos a - c), and therefore (a' + c' + k^ - 2ac cos (f) -~ = 2ga (cos

' a; = _ — integrating we get \ij r - a 2 \h sin a (cos- Q - cos^ a) /'2\i i / { - f a^ log (cos B + (cos^ B - cos" af] = + C ■ \f^J sin a but B = a when t = ; hence C = I - y ct^ log cos a ; and therefore 2\i 2 1 cos B + (cos'- B - cos" of t cr loo- ^jU/ cos o sin a AYhen AC A' becomes vertical, B = 0, and we have for the re- quired value of t, 2\l 3 • 1 1 + sin a /2V o . 1 , TT + 2a - I a sin a log = i - a" s\n a log tan . uj cos a \iuj 4 (7) BFG (fig. 193) is a heavy body of any form, of which C is the centre of gravity ; an inextensible string attached to a fixed point F is wound about the circumference of a circle ALU, having C for its centre, and representing an axis ; LA is vertical ; to determine the velocity of C when the body has descended from rest through a given altitude, under the action of gravity, by the uncoiling of the string. Let a be the radius of the axis, k the radius of gyration of the body about C: v the velocity acquired by C, after descending through a space x. Then 2ga:x a +K This problem is one of the ' Theoremata Selecta,' given by John Bernoulli, 'pro conservatione virium vivarum demonstranda et cxperimentis conjirmanda. Comment. Acad. Petroj). 1727, p. 200. Opera, tom. iii. p. 127. DYNAMICAL PRINCIPLES. 397 (8) A particle A (fig. 194) descends down the curve CKA, drawing a particle B up the curve CLB by means of a string passing over the point C; to determine the velocities of the particles after moving from rest through any corresponding spaces. Let m, m', be the masses of A, B, respectively ; v, v' , their velocities after moving through vertical spaces equal to y, y ; then, ds, ds' , denoting elements of the two curves, w v" ds " «' = 2<7 {my - 1)1 y) - -~^ -—^ , v'' = 2g {my - m'y') tiuW + mlds'"^ ' if J J J mds' + m'ds'"^ John Bernoulli; Act Erudit. Lips. 1735. Mai. p. 210 ; Opera, tom. iii. p. 257. Hermann; Memoires de St. Petershourg, tom. ii. D'Alembert ; Traite de Dynamique, p. 123 ; Seconde Edition. Sect. 2. Vis Viva mid the Conser'cation of the 3Iotion of the Centre of Gravity. The Principle of the Conservation of the Motion of the Centre of Gra\dty, under its most general form, asserts that, the motion of the centre of gravity of a free system of bodies disposed rela- titely to each other in any conceivable manner, is ahcays the same as if the bodies tvere all united in the centre of gravity, and at the same time each of them were animated by the same accelerating forces as in their actual state. The discovery of the Principle is due to Newton,^ by whom it received a demonstration in the particular case whei-e the system is subject to no external force, when the centre of gravity will either remain at rest or move in a straight line with a uniform velocity. D'Alembert" afterwards extended the Principle to the case where each body is supposed to be solicited by a constant accelerating force acting in parallel lines, or directed towards a fixed point and varying as the dis- tance. Finally, Lagrange' expressed the Principle under its most general form for every law of force to which the bodies can be subject. ' Principia ; Axioviata sive Leges Motus, Cor. 4. * Traite de Dynamique, Seconde Partie, Chap. ii. ' Meeanique Anahjtiq'ie, tom. i. p. 257, &c. 398 DYNAMICAL PKIXCIPLES. (1) A smooth groove KAL (fig. 195) is carved in a vertical plane in the body KBCL, which is placed upon a smooth hori- zontal plane, along which it is able to slide freely ; to find the form of the groove that a heavy particle, placed within it, may oscillate in it tautochronously, the time of an oscillation being given. Let P be the place of the particle in the groove at any time ; draw PN vertically to meet the horizontal plane in N, which will lie in the line OE formed by the intersection of a vertical plane through the groove with the horizontal pkne. Let A be the lowest point of the groove, draw AM horizontally, AA! ver- tically. Let be a fixed point in OE; OA' = x, ON=x^, PN=y^, AM=x, PM=y', let h^, k, be the initial values of y^,y ; m = the mass of the particle, m! = the mass of the body. Then, by the Principle of the Conservation of the ]\Iotion of the Centre of Gravity, since no forces act upon the particle and body parallel to OE, ,dx dx. ^ .. m — - + VI —r-^ (1). dt dt ^ ^ Also, by the Princij^le of the Conservation of Vis Viva, , dx'- (dx^ dy?-\ ,, , ,^^ But, from the geometry, it is evident that dx^ _ dx dx 'df~dt'^dt ^' ™'^ ^'-^'-'-y- '^-% w- From (1) and (3) we have dx^ m dx dx m dx . . dt m + wi' dt ' dt m + 711 dt ' ' ' ' Hence, from (2), (4), (5), we see that n't dx' dir ,, . m DYNAMICAL PRINCIPLES. 399 and therefore, if t denote the time of a semi-oscillation. rof m dx~ \i (2(/f ' ^ (Ic- yj This value of t must be independent of k in order that the particle may oscillate tautochronoiisly, and therefore we must have, it being necessary that the coefficient of dy be of - 1 dimensions in y and k, ( m' ^a.\ \x^a ^^^^ \m + rd dy , y where a is a constant quantity ; hence dx (m + m\ (cL - y\ dy \ 'id J \ y and therefore, bv intea^ration, But, from (6) and (7), 1 r° ady _ ira ^ _ 2^ {'^gfjAJ^y-yi C^gf' and therefore from (8) we get, for the equation to the groove, X = ; — r I -^ y - y- - + '^ vers'^ -4 • Clairaut ; Memoires de V Academie des Sciences de Paris, 1742, p. 41. Euler; Ojitiscula, de motu corporum tuhis mohilibus inclusorum, p. 48. Sect. 3. Vis Viva and the Conservation of Areas. The Principle of the Conservation of Areas asserts, that if a system of particles he subject only to mutual actions, the sum of the products of the mass of each particle into the projection {on any proposed plane) of the area described by its radius vector round any assigned point, is proportional to the time. The same 400 DYNAIMICAI, TRINCIPLES. principle holds good also if the system be subject to external forces, provided that they be such that the algebraical sum of their moments about a line through the assigned point at right angles to the proposed plane be zero. This principle, which is in fact a generalization of Newton's theorem respecting the areas described by a single body about a centre of force, was discovered, about the same time, by Euler,^ Daniel Bernoulli,'* and D'Arcy f the enunciation of the Principle given by Euler and Bernoulli being expressed under a form somewhat dif- ferent from that given by D'Arcy, under which it is now generally expressed. The discovery of the Principle was sug- gested to these three mathematicians by the consideration of the problem of the motion of several bodies within a tube of given form, moving about a fixed point. (1) P, n, (fig. 196), are two material particles attached to an infiexible straight line POT[, moveable in a horizontal plane about a fixed point ; the particle n is fixed to the inflexible line, while the particle P is capable of sliding along it ; to de- termine the path described by P, corresponding to any initial velocities of the particles. Let OE be an immoveable straight line passing through ; PO = r, UO - a, m = the mass of P, />t = the mass of IT, Z POE = 0. Then, by the Principle of the Conservation of Areas, since the only force to which the moving system is subject is the reaction of the fixed point O, we have {mr + ^ci') -j^ = C (1), where C is some constant quantity. Again, by the Principle of the Conservation of Vis Viva, ""idf^' Tf)^^''dF = '^' C being a constant quantity. ' Opuscula, de motti corporum tubis mohilibus inclusorum, p. 48, 1746. ^ Memoires de I'Academie des Sciences de Berlin, 1745, p. 54. ' Memoires de I'Academie des Sciences de Paris, 1747, p. 348. DYNAMICAL PRINCIPLES. 401 Eliminating dt between (1) and (2), we obtain dr ^ c C' 1 which is the differential equation to P's path. In order to determine C and C , suppose that h, w, u, are the dB dr initial values of r, -^ , -r-, respectively. Then, from (l\ dt dt (;mb^ + fji.a') w = C, which determines C; and, from (2), mu^ ■+ {mh^ + iMa~) w^ = C , which determines C". Clairaut; Jfe'm. de VAcad. des Sciences de Paris, 1742, p. 22. D'Arcy; Mem. de VAcad. des Sciences de Pam, 1747, p. 351. D'Alembert; Traite de Dy- namique, p. 104, seconde edit. (2) A straight rod PQ, (fig. 197), subject to the condition of always passing through a small fixed ring at O, is in motion on a horizontal plane; to determine the path of its centre of gravity G. At any time t of the motion let OG = r, L GOE= 0, OE being a fixed line in the plane. Let m be the mass of an element of the rod at any distance p from O, and let ij. be the mass of the whole rod. Then, by the Principle of the Conservation of Areas, the only force which acts on the rod being the reaction of the ring, k being the radius of gyration of the rod about its centre of gravity, and C a constant quantity. Again, by the Principle of the Conservation of Vis Viva, the ring being considered perfectly smooth, DD 402 DYNAMICAI, PRINCIPLES. dr' , , ,„ dO' ,^, ^^df^'-^'^'-^de ^'^' C being a constant quantity. Eliminating dt between (1) and (2), we have = {^Cr^-^/t^)-lj(r^ + /5.^).... (3). dr^ dO' Suppose that a, u, w, are the initial values of r, — , — , re- at civ spectively; then, by (1) and (2), hence the equation (3) becomes which is the differential equation to the path of 6r. Clairaut ; Memoires de I' Acad, des Sciences de Paris, 1742, p. 38-41. (3) Two equal particles P, P, (fig. 198), are attached to the extremities of a rod PP ; the middle point of the rod is fixed ; the rod is able to move in every direction about O ; to determine the motion of the particles corresponding to any initial circumstances, the weight of the rod being neglected. Through the point draw a straight line A OB ; with as a centre and radius equal to OP, describe the two indefinite cir- cular arcs APk, Al, the latter of which is supposed to lie within an assigned plane. Let OP=a, LAOP=(p, LkAl=B; Wi = the mass of each of the particles. Then, t denoting the correspond- ing time, we shall have, by the Principle of the Conservation of Vis Viva, whether the particles be subject to the action of gravity or not. DYNAMICAL PRINCIPLES. 403 ydtr ^ (It- J C, c, being constant quantities. Again, by the Principle of the Conservation of Areas, we have dO 2ma' sin^ d J But, the strings being supposed to be kept at full stretch, we have r + r' = c, r + r" = c, (2) ; where c , c , denote the lengths of the strings mOm', mOm' ; and therefore d\ dV _ d'r dV 'df^'df~' If^lf'' hence, by the first and third of the formulae (1), T' T + T de" , dO" ,,, — r + = r —^ + r -— (3) ; ni 771 df df ^ and, by the second and thii-d. T T + T de" „ dO" — 7, + = T h r — m m df df -yr + r -YH- • • (4)- Again, since the only forces which act upon the three particles pass through the point O, we have, by the Principle of the Con- servation of Areas, 2 d9 „ dd' , .., dO" „ dt (5); dt dt where e, e , e" , are invariable quantities : hence, from (3) and (4), have T T + T" — + in 1) = -T. + V3 ' T_ T + T" _e^ e^ _ m 7n r^ r^ ' from these two equations we may readily ascertain that m e ■'3 ' , , „. T me' (m + m) e" (m + m + m) — r = — v h tt^ — , , „. T me^ im + wi') e"^ me'"^ {■m + m + m) ^ = —ir- + ,., jrr- , ....(6), (;;? + ni + ni) „, T + T (m + m) e' me' m e m, DYNAMICAL PRINCIPLES. 407 ■which give the values of the tensions of the two strings mOm' , mOm, and of the double string Om. It is important to observe that these values for the tensions hold good only so long as both the strings are at full stretch ; if either of the strings become slack at any epoch of the motion, these formulae will no longer apply ; this will be evident when it is considered that in obtaining them we made use of the equations (2) which are grounded on the supposition that the strings are at fall stretch. The formulae themselves will indicate the epoch at which theh iuapphcability may commence by giving a zero value for either T' or T'. Again, by the Principle of Vis Viva, / , de^ dr'\ , ( „ d^"" dr"\ , / ,. dB" dr"\ ^ \ df df J \ dtr dt' ) \ df dfj where C is some constant quantity : hence, observing that, by dv dv dv the equations (2)^ — = - — = — ^ we get iXh etc etc , dO' , ,2 dB" „ ,„ dB" , , „. dr" ^ df df df df and therefore, by the equations (5), me^ me"^ rri'd'^ , , „. dr^ &" ^ ,^. ^r + -7^ + —r;r + (^ + ^* + ^^ ) 77^ -4 = ^ (7)' r' r r clu t and thence, by the equations (2), putting m + m + in = /x, me^ m'e'^ m"e"* //e^ dr^ . r (c - r)- (c - r) r* da which is the differential equation to the path of ?n. Similarly may be obtained the differential equations to the paths of m and m. These equations will evidently cease to define the paths of the particles if at any time either of the strings become slack, or either T' or T become zero. If either of the strings become slack at any time, then we shall have to investigate the motions of the two particles whose connecting string is not slack, the par- ticle which belongs to the loose string moving along for a time without constraint. From the equation (7) it is e^ddent that none of the quantities r, r, r , can ever become zero; or that the particles, so long as the strings are tight, will none of them arrive at the point O. 408 DYNAMICAL PRINCIPLES. Suppose that w, J, J, a, (5, are the initial values of — , -=- , cic ecu -— , r,-r- ; then, from the equations (5), at at e = a^w, e = (c - a)'w', e' = (c" - of J' ; which give the values of e, e, e" : and then, from (8), C = mc^ta + rd (d - of J' + m" (c" - of J'' + ^j3^ If instead of attaching two particles m', w", to m, we had attached any number of them, the problem would have been essentially of no greater difficulty. Riccati; Comment. Bonon. torn. v. P. i. p. 150; anno 1767. (6) Two particles P,P', (fig. 200), are connected together by a rigid rod without inertia which passes through a small smooth ring at ; the rod rests upon a horizontal plane : supposing any impulse whatever to have been communicated to the particles, to find the paths which they will describe. Let OE be a fixed line in the plane of the motion ; OP = r, PP = I; a, a', the initial values of OP, OP' ; m, m! , the masses (10 dr oiP,P; lPOE^Q; io,Q, the initial values of -f , ^ . Then at at the differential equation to P's path will be {mr^ + m' (I - r)~] {A [mr + m' (I - rf] - 1 } = (m + ?n") -j^, , , . (m + ni) (y + (ma' + m'a'^) u>^ where A = -. 5 — - — , ,.,, ; {ma" + ma') w and similarly for the path of P. Clairaut; 3Iem. Acad. Paris, 1742, p. 38. D'Arcy ; lb. 1747, p. 352. ( 409 ) CHAPTER XI. COEXISTENCE OF SMALL OSCILLATIONS. Conceive that a particle or a system of particles, subject to cer- tain fixed laws of geometrical connection or constraint, be slightly but generally deranged from a position of stable equilibrium, the invariable elements of the geometry being supposed to be free from particular relations. Then, if in the geometrical equa- tions there be n independent variables, the motion of each member of the system may be represented by the composition of )i primary oscillations of dififerent periods, the periods of the n oscillations of any two members of the system being coexistent, while their amplitudes will generally be diiferent. When the periods of the 7i elementary oscillations are commensurable, the whole system will return to its original state after an interval equal to the least common multiple of these periods ; as in the case of vibrating cords and vibrating surfaces. This general property of sympathetic vibrations has been entitled the Princi- ple of the Coexistence of small Oscillations or Vibrations. Should the original derangement of the system from its posi- tion of equilibrium, instead of being perfectly general, be effected by peculiar adaptation, we may reduce the w elementary oscilla- tions to any smaller number we may please. If the fixed geometrical elements of the system be not, as we have supposed, free from particular relations, and if it receive a perfectly general derangement, there will as before arise in the system altogether n classes of oscillations ; under these circum- stances however a peculiarity occasionally presents itself, viz. that, although as we have supposed the original derangement be quite general, yet into the motion of no single member of the system will all the elementary oscillations enter ; this case will then constitute a failure of the Principle of the Coexistence of small Oscillations. The Principle of Coexistent Oscillations was first laid down 410 COEXISTENCE OF SMALL OSCILLATIONS. by Daniel Bernoulli, who has written several memoirs on the subject in the St. Petersburgh Transactions. See particularly Nov. Comment. Petrop. vol. xix. p. 281. The student is referred also to Lagrange, Mecanique Analytique, torn. i. p. 347, and to Poisson, Traits de Mecanique, torn. ii. p. 426, where he will find investigations of the Principle based on the first principles of Mechanics. (1) To determine the nature of the oscillations of a material particle within the surface of an ellipsoid, in the neighbourhood of the lower extremity of a vertical axis. Let 2a, 25, denote the lengths of the two horizontal axes of the ellipsoid, 2c representing the length of the vertical one ; and let the co-ordinate axes be so chosen that a, h, may be parallel to the axes of x, y, and that c may coincide with the axis of z. Then, by D'Alembert's Principle combined with the Principle of Virtual Velocities, we have for the motion of the particle, d'^x ^ d^xi ^ fd'z \ rv ^ .X w^^^i^^y^yie^^Y'-" «• where x, y, z, denote the co-ordinates of the particle at any time t, and Sa:, Sy, Zz, the increments of x, y, z, in passing to any point of the surface very near to the position of the particle. Again, by the equation to the ellipsoid, we have a b c and therefore, neglecting powers of the small quantities beyond the second, \^t-t}f = c(\- — -^ a' h- 7 V 2a' 2b X y a b zz=c(^^yh\ hence, from (1), neglecting the products and powers, beyond the first, of small quantities in the coefficients of ^x, Sy, we get COEXISTENCE OF SMALL OSCILLATIONS. 411 and therefore (^ +| ;,) g^ + (g ^ ff j,) gy = 0. Equating to zero the coefficients of Sx, Sy, which are indepen- dent of each other, we get S-?-« ••••(^)' §^l^;» «■ The integral of the equation (2) is I « J and that of (3) is . f(caf , J y = 7 sm ^-j- i + ^ji where j3, y, e, 4 , are arbitrary constants, which may be determined from the initial values of x,y,-j- , —-. It may be observed that the oscillation of the particle depends upon two simple oscilla- tions of which , , are the periods ; the number of inde- pendent simple oscillations being the same as the number of independent variables in the geometrical equation to which the position of the particle is subject. Poisson ; Traite de Mecanique, tom. ii. p. 439. (2) A uniform rod AB, (fig. 201), which is connected by a string OA with a fixed point 0, having been slightly displaced from its position of equihbrium ; to investigate the nature of its small oscillations. Draw vertically the indefinite straight line Ox ; take P any point in AB, draw PM at right angles to Ox, and produce BA to meet Ox in C. Let AB = 2a, 0M= x, PM= y, AP = s, 0A = l, lAOx = B, lBCx = ((>. Then for the motion of the rod we have, by D'Alembert's Principle combined with the Principle of Virtual Velocities, 412 COEXISTENCE OF SMALL OSCILLATIONS. where dz^ dy, denote the small spaces described by the element ds of the rod in the time dt, parallel to the co-ordinate axes ; ^z, ^y, denoting the resolved parts of its virtual velocity. Now, fi-om the geometry, we have z = I cos + 5 cos 0, y = I sin + s sin ; and therefore, our object being to transform the equation (1) into an equation invol\4ng Q, 0, instead of z, y, and to retain small quantities only as far as the first order in the coefficients W, ^, d^z d'^y , d'B d'ch df ' df df df hence, substituting these valiies of a:, y, . . . . in the equation (1), we have J ""{yds {10 BO + s0 g0)} +p S.ds U^ + s ^) (no + s d...), , = /3sin{(0.. 414 COEXISTENCE OF SMALL OSCILLATIONS. and the oscillations of B and ^ will evidently be regular and — If a, j3', be not equal to zero, the oscillations of B and (p will be compounded of two simple and isochronous vibrations. Suppose that at two diiFerent times t', f, the values of B and of -rr- are the same. This will manifestly be the case if dt •' \m'J m 9 and ( ^, r ^"+ ^'= Ly'/ ^' + ^' + 2A'7r, X, X', being any integers ; hence and therefore Xw* = X'w'*, or m, w', must be to each other as two square numbers. It will be observed that, in agreement with the general theory of the Coexistence of small Oscillations, the number of independent oscillations of B and is two, which is the same as the number of the independent geometrical variables. The following is another method of soMng this problem. Let G (fig. 202) be the position of the centre of gravity of the rod at any time t; draw GH at right angles to the vertical line Ox; m =^ the mass of the rod, mk^ = its moment of inertia about G, T= the tension of the string AO, OH==x, GH=y. Then, the rest of the notation being the same as before, we have, for the motion of the rod, m^ = - TsinB (2), mk' ^ =- asm(,}>-B)T,... (3). COEXISTENCE OF SMALL OSCILLATIONS. 415 Eliminating T between (1) and (2), and omitting small quan- tities of higher orders than the first, we have '-ff-^'-' w^ and, eliminating T between (1) and (3), we get in the same manner ,72. ^^^ + «^(0-0)=O (5). But y = -e) .... (3). at From (1) and (2) we get, as far as small quantities of the first order, ^2,, nf^^'-" «' and, from (l) and (3), to the same degree of approximation. Now, from the geometry, y -(a - h) sin B -v c sin = (a - 5) + c0, nearly : hence from (4) we obtain («_5)^^ + c^ + ^0=O (6). ^ ^ dir df -^ ^ Assume = a sin i ( ^ V ^ + £ i , tp = ^ smU^J t + A'. then from (5) we may get ^(icr-k') = acr (7), and, from (6), ^jS = a [r - {a - h)}, and therefore, eliminating a and jS, {cr - k~) (r - a + b) = c^r. Let the two roots of this quadratic in r be denoted by m and m' ; then, for the general values of B and (j>, we have COEXISTENCE OF SMALL OSCILLATIONS. 417 e = a sin ff^y t+s\+a' sin l.f^]' t + A.... (8), ,=/3sin0J...}./3'si„|(l)'..ej....(9> From (7) we have, /3, /3', being the values of /3, and a, a, those of a, corresponding to the values m, m', of r, r, acni r,, a'cni ^ = T^ ' ^ =^ —' 7^ ' cwi - a; cm - X; hence, from (9), we have = ^ sm ^ M- f ^ + 6 U — ; — p sin ■ M^ W + e' ^ . . (10). cm - k iynj J cm - k: \\m J J In the equations (8) and (10) there are four arbitrary con- stants, a, a', g, i , which may be determined if we have given the initial values of 0, rf», -— , -^ . ' ^' dt' dt If a =0, j3' = 0, we have » = ° ="{©■*-}> *^''^'"{©''-]^ and the oscillations of and will be regular and isochronous, the time of vibration being tt [ — If a and /3' have finite values, the oscillations of and will be compounded of two simple isochronous oscillations. Euler; Acta Acad. Petroj). 1780, P. ii. p. 133. (4) A string AEFB (fig. 204) is attached to two fixed points A, B, in the same horizontal line. From E, F, points so chosen that AE, EF, FB, are all equal, two masses are sus- pended by strings EM, FN, of dififerent lengths, the masses being equal. Supposing the system to be slightly deranged from its position of equilibrium, to investigate the nature of its small oscillations. At any time t let EM, FN, make angles ^, 0', with the vertical. Let AE, EF, BF, make angles a + w, w, a - J', with the horizon, the values of these angles being a, 0, a, for the position of equilibrium. Draw 3Im, Nn, horizontally to meet the vertical line Amn in the points ?n, n. Let AE=EF=FB=a, EM=h, FN= U, Am = x, Mm = y, An = x , Nn = y . KE 418 COEXISTENCE OF SMALL OSCILLATIONS. By D'Alembert's Principle and the Principle of Virtual Ve- locitieSj we have, for the motion of the system, Our object is now to express x, y, x , y , in terms of w, <^, (p', and to substitute their values in this equation. This com- putation must be effected as far as small quantities of the second order. By the geometry it is plain that a cos (a + (o) + a cos to' + a cos (a - w") = 2a cos a + a, and therefore cos a(l-| w")-sin a.w+l-g w'"+C0Sa(l-2W^)+sina.t.;=2C0Sa+l; whence cos a .w^ + 2 sin a . w + w'^ + COS a . w'^ - 2 sin a . w' = ...(2). Again, by the geometry, a sin (a + (1))= a sin J + a sin (a - w"), and therefore sin a (1 - 1 w') + cos a . w = w' + sin a (1 - ^ w"^j - cos a .' = a cos a (1 - 1 w*) - a sin a . w + a (1 - g w'") + Jeep' , St/' = Jc'h' + ia sin a -— - k -~ 1 (a sin a ^w - Aci^) = 0. A; — tV - «! sm o —— [k dm - a sm a ow ) df dtj ^ EE 2 420 COEXISTEXCE OF SMALL OSCILLATIONS. Hence, equating to zero the coefficients of the independent quantities ^, Iq,', dw, we get df ^ ^1^^ " ^ + 5^0 = (5), (^^w cP(j) iP(h' 2 + 4 cos' a „ .^. 2a -— - A; sm a — ^ - A; sm a -4- + (7 .- w = 0. . . .(7). df dt- cW -^ sin a ^ Eliminating --| and — ? between (5), (6), (7), we get 72 2a sin a cos^ a -p + . d'lM) k —4- - « sin a -— ^ + (/d) = 0, which are the same thi-ee linear equations as (5), (6), (8), in the former solution. If k be equal to k! , the cubic equation (9) of the former solution will degenerate into a quadratic, and the variations of w, 0, ^', will no longer be expressible by the composition of the same elementary vibrations. This will be an instance of the failure of the Principle of the Coexistence of small Oscillations. Euler; Act. Acad. Petrop. 1779, P. n. p. 95. COEXISTENCE OF SMALL OSCILLATIONS. 423 (5) One of the scales of a common balance having been slightly displaced from its position of rest, in a vertical plane passing through the beam; to investigate the nature of the oscillatory motions of the two scales and of the beam, to which the displace- ment will give rise. Let (fig. 205) be the point of suspension of the whole balance, G its centre of gravity, AB the beam, P and Q the scales, wliich are here supposed to be material points. Draw aOb horizontal, aAa, hB^, vertical. Let AC=^a = BC, OC=b, OG = c, AP = 1= BP, 3Ik- = the moment of inertia of the beam about 0, m = the mass of P and of Q supposed to be equal. Let

" = 0. Then, bearing in mind that, initially, 0=0, »? = f, 0=0, d^ c/t/ dS dt ' dt ' dt where c is a known constant, we shall obtain for the complete expression of the motions ^, t\, v^, t\, R; and in like manner for the latter body we shall have six equations involving w/, w^', w,'j «,', v^', v^, R. Thus we shall have in all twelve equations involving thirteen variables. Another equation is supplied by the condition that the points of the two bodies in which their contact takes place shall have an equal resolved velocity in the direction of the common normal. Thus we shall be able to determine completely the modification of the motions of the two bodies due to the force of compression as well as the magnitude 426 IMPULSIVE FORCES. of this force. An aclditional modification must be applied, in the case of elastic bodies, in consequence of the force of restitu- tion eH, which, from the investigation for the former stage of the collision, has become a known force. If one of the bodies be immoveable, the simplification of the method of investigation which we have described is obvious, the thirteen equations of which we made mention being reduced in this case to seven, and the common normal velocity of the 1 vo points of contact being zero. For ample information on this subject the student is referred to Poisson's Traite de Mecanique, tom. ii. p. 254, seconde edition. Sect. 1. Single Body. Stnooth Surfaces. (1) A beam of imperfect elasticity, moving anyhow iii a vertical plane, impinges upon a smooth horizontal plane ; to determine the initial motion of the beam after impact. AVe will commence with supposing the beam to be inelastic ; in this case the extremity of the beam which strikes the ho- rizontal plane will continue after impact to slide along it with- out detaching itself. Let PQ (fig. 207) represent the beam at any time after collision ; KL being the section of the horizontal plane made by the vertical plane through PQ; G the centre of gravity of PQ; draw GH at right angles to KL. Let GH=y, QG = a, lGQH=B; Z; = the radius of gyration about G, m = the mass of the beam ; w, J , the angiilar velocities of the beam about G estimated in the direction of the arrows in the figure, just before and just after impact; u, v, the vertical velocities of G estimated downwards just before and just after impact ; B the blow of collision. Then, w' - w being the angular velocity communicated by the blow, we shall have, if (5 be the value of 6 at the instant of i^^P^^ct, ^ ^^ cos jS w — ai = (1); and, u - V being the velocity of G which is destroyed by the blow, ^ u - V = — (2). m IMPULSIVE FORCES. 427 Again, by the geometry, we get y = a sin Q ; and therefore, t denoting the interval between the instant of collision and the arrival of the beam at the position represented in the figure, ^ ^jq -r- = a cos V —- ; dt dt hence - v, - J , beinsf the values of — , -r- , at the instant after ° dt dt collision, we have ^ = « cos ^.J (3). From (1), (2), (3), we get B ^ f Ba cos i3\ u = a cos p f w + ,, , — 1 + r^ cos \i\ = u- aio cos p, -r, j^ u - aix) cos 3 /.\ jB = m^- -2 r75 — r^ ^"*)- a cos p + A; Hence, from (1), , ryU - aio cos j3 au cos /3 + k'w (O = W + a cos P -2 2~0 7^ = 2 WJ 7^ ' '^ a^ cos^ (i + h' or cos^ ^ + ^' and, from (2), ,, u - CTw cos 3 o '^^^ cos 3 + ^"w v = u-F -5 2-^ — 7^ = a cos 3 —3 ^'-y. — - . a* cos' 3 + ^' «' cos' 3 + ^ Next let us suppose the beam to be imperfectly elastic, its elasticity being denoted by e ; in this case the value of B given in (4) must be increased in the ratio of 1 + e to 1 ; and therefore, instead of the equation (4), we have ^ , , ,, u — au) cos 3 B = {\ +e) mTe -^ 3^5—70 , a cos p^-k which determines the magnitude of the blow of impact : substi- tuting this value of B in (1), we get u - ai^ cos 3 w' = w + (1 + e) a cos 3 a^ cos' 3 + ^^ {k' - a^e cos' ^) w + {\ + e) au cos 3 «' cos' 3 + ^^ 428 IMPULSIVE FORCES. and, substituting in (2), /, A 72 u - au) cos (3 v = ti-(l + e)k' -^ ^-^ ^ a cos" p + fc a'u cos^ [5 - ek'u + (1 + e) k^aia cos /3 d cos^ ^ + k^ The velocity of G parallel to the plane KL will be the same before and after impact. The end B of the beam ■nill evidently after collision detach itself from the horizontal plane, since v is less and J greater when e has a finite value than when it is equal to zero. (2) A body AB, (fig. 208), after sliding from a given altitude down an inclined plane Oy, impinges against a small obstacle at C; to determine the impulsive reaction of the obstacle and the motion of the body immediately after collision. Let G be the centre of gravity of the body ; draw GH at right angles to Oy ; Ox parallel to HG. Let GH - a, CH = b, m = the mass of the body, k = the radius of gyration about G', c = the velocity of G immediately before impact. We will commence with supposing the body to be perfectly inelastic ; in this case the point C of the body will remain dui'ing collision in contact with the obstacle, the body rotating about this point. Let R, S, denote the impulsive reactions of the obstacle parallel to Ox, yO; and let u, v, denote the velocities of G parallel to Ox, Oy, on the completion of the impact ; also let w represent the angular velocit^^ of rotation about G at the same instant. Then Ave have, for the motion of translation, mu = M (1), fnv = mc - S (2) ; and, for the motion of rotation, rnk'^oj = Sa- Rh (3). Again, the velocity of the point C of the body, estimated parallel to Ox, will be It - ioCG cos L GCH or u - hw, the former term of this expression arising from the motion of G, and the latter from the rotation of the bodv about G. IMPULSIVE FORCES. 429 Also the velocity of the point C, parallel to Oy, will be V - M .CG sin L GCH or v - aio, the former term being due to the motion of G, and the latter to the rotation about G. But the point C of the body, which is perfectly inelastic, remains at rest during the collision ; hence, e\ddently, u - bu) =^ (4) ; V - aw = (5). From (1) and (4) we have M = mbu) (6), and from (2), (5), S = m (c - acj) (7) ; substituting these values of M and ^S* in (3), we obtain k'(o = ac — ci'u) — b'u), and therefore, ac abc a^c u - ""- a'^-b' + k'' a' + h'^J^' aUb' + k' hence also, from (6), „ mabc i^ + h' + H ' and, from (7), <^- ( g'c \ _ mc (V +_k') '^ V'' ~ a' + b' + W) ~ a' + bU F • If the body be supposed to be elastic, we must increase these values of R and S in the ratio of 1 + e to 1, e denotmg the elasticity. Hence -r. _ m (\ + e) abc ^ mc (l + e) {b' + k') _ a + b + k a + b ^ k and therefore from (1), (2), (3), _ (1 + e) abc _ d^ - e{b^ -vT^) _ (l + e) ac "" ^ aUb' + k' '''"'' ~^^~T"5^T1'-^ ' *" ~ a' + b' + ¥ ' (3) A beam AB (fig. 209) is originally in a vertical position, hanging from the point 0, along the line Oij ; supposing the extremity A of the beam to be projected from O with a given velocity along a smooth horizontal groove Oz, to determine the motion of the beam. 430 IMPULSIVE FORCES. Let AB be the position of the beam after a time t from the projection of A, G its centre of gravity ; draw GH at right angles to Ox; let OH = x, GH=y, L OAG =6, AG = a ; m = the mass of the beam, k = its radius of gyration about G. Then, for the motion of the beam at any time after the pro- jection, we have, by the Principle of the Conservation of Vis ' fdx^ dtr J. d6'\ ^ ^ ,,. and, by the Principle of the Conservation of the Motion of the Centre of Gravity, — = C (2). dt ^ ^ From (1) and (2), we have but, from the geometry, we see that y = « sin Q ; hence m (a^ cos^ + k') -p- = C" + 27nga sin (3). Let B denote the blow of projection which is impressed upon the end A of the beam; u the velocity of ^'s projection, and w the angular velocity of the beam about G immediately after the blow. Also let V be the velocity communicated to G by the blow. Then we shall have mv = B, mk'w = Ba, (4). Agam, the velocity of A along Ox will be equal to V + aw, the former term being due to the motion of G, and the latter to the rotation about G ; but the velocity of A is also u by the hypothesis ; hence v + cio, = u; but, from the equations (4), we have k'u) = av; we obtain, there- k' a" + k^ a- + k^ IMPULSIVE FORCES. 431 JO Now, S = Iv, — = w, simultaneously; hence, from (3), mk^t/ = C" + 2mga ; and tlierefore (a^ cos^ 9 + k~) -— =r kW - 2ga (1 - sin 0) '"'''' -2^a(l -sin0)...(5). (a- + kj Also, the value of — beinar constant, as is shewn by the dt equation (2), dx _ k\i k'ut di~^^ ~crni ' "^^ a^WW ' which gives the velocity of G parallel to Ox, and the value of x at any time of the motion ; the angular velocity of the beam for every position is given by (5), whence 9 is to be ascertained in terms of t. (4) An inelastic beam AB, (fig. 2 1 0), capable of moving in a vertical plane about a fixed horizontal axis thi'ough A, falls from a given position, and impinges against an immoveable obstacle at C; to determine the shock on the axis. Let G be the centre of gravity of the beam ; AM a horizontal line through A; m = the mass of the beam ; L GAM = at any time t of the descent ; a = the initial value of ; k = the radius of gyration about G ; AG - a. Then, for the motion of the beam in its fall, m {c^ + M) -— = mga cos ; multiplying by 2 — and integrating m {a' + U) —J- = 2macj sin + C ; 7/1 but = a when -— = : hence dt - Imag sin a + C, and therefore (a^ + W') -j^ = 2ag (sin B - sin a). 432 IMPULSIVE FORCES. Let L CAM = j3 and, at the instant before collision, let — = w ; then, clearly, dt (a* + W') w' = lag (sin /3 - sin a) (1). Let It, m , denote the impulsive reactions of the obstacle C and the axis A, at the instant of impact ; both of which will evidently be at right angles to the length of the beam. Now the effect of the reaction It is to destroy the whole of the angular velocity of the beam about A, by impressing upon it an equal and opposite angular velocity ; hence, putting CA = c. mu, (a" + k-) = Be (2). Again, the difference of the moments of It and It' about the centre of gravity of the beam being R(c ~ a) - B!a, yve must have R{c - a) - Ma = mk'w (3). From (2) and (3) we obtain mo> (a^ + k') (c - a) - H'ac = tiik^cu/ M'ac = mu> {(c - a) (a^ + k^) - ck^}, 2 7 2^ It = niio ' a > ; L C J and therefore, from (1), 73' /r, Nr/sin B - Sill a\ ( <£' -V U If M' = 0, we must have a" + k'^ ^ a' + W a = 0, c = — - - ; c a and therefore C must be the centre of oscillation of the beam at the moment of collision. If the beam be elastic we must increase the value of R given by (2) in the ratio of 1 + e to 1, e denoting the elasticity; we shall then have, from (3), B = — {{\ ^ e){c- a) {(i + m - cm. ac "* IMPULSIVE FORCES. 433 (5) An inelastic beam, which is moving without rotation along a smooth horizontal plane, impinges upon a fixed rod at right angles to the plane ; to determine the impulsive reaction of the rod and the motion of the beam subsequent to the impact. Let AB (fig. 21 1) be the position of the beam at the instant of impact ; O the place of the obstacle ; G the centre of gravity of the beam; G'G the line of G's motion before collision. Pro- duce OB indefinitely to x, and draw the indefinite line ]/0y' at right angles to Ox and meeting GG in G'. Let R = the im- pulsive reaction of 0, which will be exerted along the line Oy' ; u = the velocity of G before impact ; lOG'G = a ; OG = c ; h = the radius of gyration of AB about G ; m = the mass of the beam; v^, v , the velocities of G parallel to Ox, Oy, just after impact, and w the angiilar velocity about G. Then, by the equations of impulsive motion, mv = mu sin a (l )? mv^ = mu cos a - R (2), mk'tu = Re (3). Again, the velocity of the point of the beam in the direction Oy, the instant after impact, must be v^ - cw, v^ being its velocity due to the velocity of G, and - cw its velocity due to the rotation of the beam about G ; but, the beam being inelastic, the efiect of the impact is to destroy the resolved part of O's velocity at right angles to AB ; hence v^ must be equal to Cw. We have, then, from (2), mcw = mu cos a - R, and therefore, by the aid of (3), mc'w = mcu cos a - tnk'u), cu cos a or W = -^ Y:r • c + k' Hence, from (3), _ mk'u cos a ^ = i 72—5 c + k and consequently, from (2), kSi cos a c'u cos a V = u cos a -, ,, = —. jir • ^ c- + k- c' + ¥ Y F 434 IMPULSIVE FORCES. Also, from (1), ^^=u sin a. Thus we have determined completely the instantaneous motions of the beam after the impact, and the impulsive reaction of the rod at 0. It may be ascertained that, if the original motion be precisely such as our particular figure represents it, on the consummation of the impact, the beam will detach itself from the obstacle and will then move along freely with the velocities «?^ , « , w, which we have obtained above. In fact we should find, if we were to assume the beam always to touch the obstacle, that the obstacle would have to exert a continuous attraction instead of a reaction. Sect. 2. Several Bodies. Smooth Surfaces. (1) A heavy sphere P (fig. 212) falls down from a given altitude upon a body at rest, the upper surface of which is a smooth inclined plane ; the body is capable of sliding along a smooth horizontal plane, its lower surface being flat; also the same vertical plane contains the centres of gravity both of the sphere and of the body : to determine the initial motions of the sphere and of the body, both of which are supposed to be per- fectly inelastic. Let ABH denote the section of the body made by a plane passing thi'ough its centre of gravity, AH being a line in the horizontal plane. Let V be the velocity of the sphere just before impact ; u, v, the resolved parts of its velocity after impact, perpendicular and parallel to the hypotenuse BA of the triangle BAH ; v! the velocity of the body parallel to AH after the impact ; m, ni , the respective masses of the sphere and body, and B the blow of collision ; L BAH = a. Then, for the motion of the sphere, the blow being at right angles to ^^, mu = mV cos a - B (1), V = V sin a (2) ; and, for the motion of the body, m'ti' = B sin a (3). These three equations involve four unknown quantities, u, v, IMPULSIVE FORCES. 435 ?/, B : for the solution of the problem, then, another equation will be necessary. This will be obtained by the consideration that, the sphere and the body being both perfectly inelastic, the effect of their collision is merely to prevent the penetration of the one into the interior of the other, without causing any recoil, which could result only from the existence of elasticity : hence the velocity of the ball, after collision, at right angles to the line BA must be equal to the velocity of any fixed point in this line estimated in the same direction. Now the velocity of any fixed point in BA at right angles to this line is evidently u' sin a ; and therefore we have u' sin a = ti (4). From the equations (1), (3), (4), we obtain mB sin^ a = imii V cos a - mB, J ^1 p T. mm' V COS a and tnereiore B = r-^ , (o), m sm a + m which gives the magnitude of the blow. From (3) and (5), we get , m V sin a cos a u = —. r , m sm a + m which determines the motion of the body ; and therefore, by (4), m V sin^ a cos a m sm a + m D'Arcy ; Memoires de VAcademie des Sciences de Paris, 1747, p. 344. (2) Two billiard balls B and C(fig. 213) are lying in contact on the table ; to find the direction in which the ball B must be struck by a tliird ball A so as to go off in a given direction BD ; the balls being of equal volume and weight, and perfectly smooth. Let the direction AB, which joins the centres of A and B at the instant of their collision, make an angle B with the straight line CBE, and let L CBD = a. We will first suppose the balls to be inelastic. Let a, a', denote the resolved parts of the velocity of ^ in the direction AB, before and after collision re- spectively ; and h, c, the velocities of B, C. Let m represent FF 2 436 IMPULSIVE FORCES. the mass of each of the balls, R the blow between A, B, and S that between B, C. Then, for the motion of B, resoh'ing forces at right angles *o ^^' mb sina = M sin d (1), and, resolving parallel to EC, mb cos a = R cos Q - S (2). Also, for the motion of C, we have mc = S (3). Again, since after collision the velocities of ^ and C in the direction EC must be equal, there is h cos a = c (4). From (3) and (4) we get mb cos a = S, and therefore, from (2), mb cos a = R cos 9 - mb cos a, R cos = 2}?ib cos a (5). From (1) and (5) there is tnb sin a cos 6 - 2mb cos a sin 0, and therefore tan 6 = h tan a, which determines the point in which A must come into collision with B. If we introduce the consideration of elasticity, the magnitudes of R and S will each have to be increased in the ratio of 1 + e to 1. Now the direction of B^s motion will e'sddently not be affected by any alteration in the absolute magnitudes of R and S, provided that the ratio between their intensities be not changed. Thus we see that the consideration of elasticity will not modify the solution of the problem. (3) A billiard ball impinges simultaneously upon two other billiard balls vrhich are resting in contact; to determine the motions of the three balls after colhsion. Let A (fig. 214) denote the centre of the impinging ball at the moment of impact ; A', -A , of the other two balls. Join AA', AA , and produce them indefinitely to points a, a ; draw Aa a common tangent to the two balls A', A". Then evidently after collision ^'s motion will be confined to the straight line Aa; while A', A", will proceed to move along IMPULSIVE FORCES. 437 A'a, A"a". Let u, v, be the velocities of A before and after impact ; v the velocity after impact of each of the balls A! , A". Since AAA!' is evidently an equilateral triangle, L aAa = 5 tt = L aAa; let B be the blo^y of collision between A, A', and A, A"; and m the mass of each of the three balls. Then mv = mu - 2B cos g tt (1), mv = B (2). Let us first suppose the three balls to be perfectly inelastic ; then the instant after impact the balls A, A', will move in contact, as well as the balls A, A"; hence we must evidently ^^^^ v = V cos I TT ; we have, therefore, from (2), B = mv cos - ; 6 and consequently, from (1), 2 TT u 2m 6 5 TT 1 + 2 cos' - 6 5 TT 3^ and therefore v' = v cos - = — w, 6 5 i J -D 3' mu and B = .rai 5 If the balls be elastic, and e denote their elasticity, we must increase the value of B in (3) in the ratio of 1 + e to 1 ; hence we have i Z'^mu , B = {l+e) -J- (4), and therefore, from (1), mv = mil -5(1 + 6) mu, t) = 5 (2 - 3e) w ; 1 ■in / N ^ X / /. N 3' miL and from (2), (4), mv = (1 + e) — - — , V = — (1 + e) u. 5 M aclaurin ; Treatise of Fluxions. D'Alembert ; Traite de Dynamique, p. 227. 438 IMPULSIVE FORCES. (4) A ball C (fig. 215) impinges upon an inelastic beam A^ with a given velocity, at right angles to its length ; to determine the magnitude of the blow and the initial cii'cumstances of the motion of the beam and ball. Let G be the centre of gravity of the beam ; m' its mass, k the radius of gyration about G ; EG = a, u = the velocity of the ball before impact, v = its velocity immediately afterwards ; B, = the magnitude of the mutual impulse : let v be the velocity of G and w the angular velocity of the beam about G just after collision. Then, for the initial motion of the ball after collision, m denoting its mass, mv = mu- R (1) ; and, for the initial motion of the beam, Wi'tj' ^ B (2), iiiTiia = B,a (3). Again, the velocity of the point E of the beam will be equal ^° ^5' + ttii), the former term of the expression being due to the motion of G, and the latter to the rotation about G ; but, the beam and ball being inelastic, the velocity of the point E of the beam after collision must be equal to that of the point E of the ball, and therefore of the point C : hence we have V = v' + a (4). From (1), (2), (3), (4), we obtain B B Ba' m m tn k and therefore u mmk^u 1 1 a^ (m + m!) J^ + mc^ m 7n mU' Hence, from (1), -jre get m 'Uu m {k^ + a^) u {m + m!) Ti + md^ {in + m') A" + mc^ ' from (2), v' = — —-j^ t. . _ 2 » mkhi (m + m') k? + ma^ IMPULSIVE FORCES. 1 /• /«\ mau and, irom (3), 439 (m + ni) J^ + ma^ (5) A cylinder is revolving with a given angular velocity- round an axis which is horizontal^ when it suddenly begins to draw up a weight, consisting of inelastic materials, by means of an inextensible string wound round the cylinder ; to determine the time the system will continue in motion, and the original distance of the weight from the cylinder, that at the instant the motion ceases the weight may just touch the cylinder. Let a = the radius of the cylinder, m = its mass, k^ = the radius of gyration about its axis; m' = the mass of the weight; let w, w', denote the angular velocities of the cylinder just before and just after beginning to draw up the weight ; u = the velocity of the weight at the commencement of its motion, B = the impul- sive force exerted initially by the string on the weight. Then we shall have Ba B w = w , t« = — ; mk m but u = aJ ; hence B Bd^ - aw - m mk^ ' and therefore _ mmaojk' mawk .,. nidr + mk ma + mk^ Let B denote the angle through which the cylinder has revolved about its axis at the end of the time t from the commencement of the raising of the weight; and let x be the corresponding distance of the weight below the horizontal plane through the axis of the cylinder. Then, by the Principle of the Conservation of Vis Viva, m -rrr + 'nik' -rr-o = 2m qx + C ( 2) : df df -^ ^ but if h denote the value of x at the commencement of the raising of the weight, it is clear from the geometry that /I 7 1 1 /> dO dx X + ad = 0, and thereiore « -^ = — ;- ; dt dt 440 IMPULSIVE FORCES. hence, from (2), rdc^ + mTi da? „ , ^ — -^r- ^='^^^-^^' dx differentiating with respect to t, and dividing by 2 — , ^ df '^ integrating Avdth respect to t, we have (m'd^ 4 mk^) — = C + mc^gt ; dx but — = - u when t= ; hence C= - (m'd^ + mk') u, or, by (1), C = - mau)Jc', and therefore {ni!a^ + mk^) — = ma^gt - maioJ? . .. . (3) ; integrating again with respect to t, we get (m'd^ + mk?) z = C + I md^gf - mawkH ; but x = b when t=Q ; hence C" = (jna? 4 mk^) h, and therefore (w'a" + m^'^J a: = {nid? + mk') h + \ ma'gf - matiJ^t ... (4). Let t' denote the time when the motion ceases for an instant; then, from (3), since — = when t = t' , dt Q = magt - mai»k , t = — ; — . m ag Hence also, from (4), since a; = when t = t', m'o'k* (m'a^ 4 mk') h = matak't' - \ m'a^gl!^ = 27n'g which gives the requii'ed value of b. (6) Two inelastic spheres, of which A and a (fig. 216) are the centres, are attached to rigid rods CA and ca, which are capable of motion in a single plane about axes through C and c at right angles to the plane ; supposing the spheres to impinge against each other with given velocities, it is required to deter- mine their initial velocities after impact. Join Aa, and produce it indefinitely both ways to points a, ft ; from C, c, draw CG, eg, at right angles to aft at the IMPULSIVE FORCES. 441 moment of collision. Let C, c, represent the lines CG, eg, il, w, the angiilar velocities of CA, ca, about C, c, respectively immediately before, and o!, J , immediately after collision, the angular motions being estimated in the directions indicated by the arrows in the figure ; B the blow of collision ; / the moment of inertia of the sphere A with its rod AC about the axis through C, i the moment of inertia of the other sphere and rod about c. Then, Ql -Q, being the angular velocity which CA gains, and w - w' that which ca loses by the shock, we shall have i(a' -q:) = b.c, i{io-w') = B.c (1). But, the spheres being inelastic, the point P of the sphere a will, the instant after collision, have the same velocity along a/3 as the point P of the sphere A : hence il'. CP . sin Z CPG = w'. cP . sin Z cPff, or Q.'C=o/c (2). Now, from the equations (l), cI{q! - il) + Ci (w - w) = 0, and therefore, by (2), we get cI(c(o' -CO.) + CH {J - w) = 0, icU+ CH) J = C(c7il + Ciui), and therefore {c^I + C'l) ^' = c (c/^ + Cio) ; which two equations give the values of O.', w'. The solution of a particular case of this problem was unsuc- cessfully attempted by John Bernoulli, son of the celebrated John Bernoulli, in the Memoires de St. Petershourg, torn, vii.: the correct solution of the problem in its most general form, was given by D'Alembert, TraiU de Dynamique, p. 221; se- conde edition. Sect. 3. Rough Surfaces. (1) An inelastic cylinder (fig. 217) having rolled down a perfectly rough plane CA, impinges upon a perfectly rough plane C'A, the axis to the cylinder being parallel to the intersection of the two planes ; to find the velocity with which the cylinder will commence its ascent up the second 442 IMPULSIVE FORCES. plane, and the limiting angle of inclination of the two planes for which the ascent is possible. Let Z CAC = a, k = the radius of gyration of the cylinder about its axis, a = the radius of the cylinder, m = its mass ; u = the velocity of the centre of a circular section of the cyhnder parallel to the plane CA just before impact, and v = the velocity after impact up the plane AC"; JR = the impulsive force of friction exerted by the plane AC upon the cyhnder at the moment of impact to secure perfect rolling. Then, for the motion of the centre of gravity of the cylinder parallel to AC, we have mv = R - mu cos a (1) ; and, for the value of the decrement of the angular velocity of the cylinder about its axis owing to the impulse R, we have the expression j^^ which, by (1), is equal to a(v + u cos o") k' ' but, the planes being both perfectly rough, it is evident that - is the angular velocity before, and - after the impact ; hence u V a (v + u cos a) ^ a a Z? a" , u - V - j^{v + u cos a) ', k but a^ = 2k' ; hence u - V = 2v + 2u cos a ; and therefore, v = 1(1 - 2 cos a) u, which gives the velocity with which the cyhnder ascends the plane AC. Since v can evidently, from the nature of the case, have neither a zero nor a negative value, it is clear that the smallest possible value of a for the ascent is given by the equation 1-2 cos a = 0, cos a = ^; whence I tt is the limiting value of a. IMPULSIVE FORCES. 443 (2) A ball sliding without rotation along a smooth horizontal plane impinges obliquely against a perfectly rough vertical plane ; to determine the subsequent motion of the ball. Let Ox, Oy, Oz, (fig. 218), be three rectangular axes, the plane xOy being horizontal and passing through the centre Cof the ball, and the plane xOz being the rough vertical plane against which the ball impinges. Let JS be the point in wliich the ball strikes against the vertical plane ; CF the direction of the motion of C before collision. Let u be the velocity of C before impact, a the inclination of CF to Ox ; v^, v , the resolved parts of the velocity of C parallel to Ox, Oy, after the impact ; w the angular velocity of the ball about C after impact ; X, Y, the im]3ulsive reactions of the rough plane along xO, and parallel to Oy, during collision ; m the mass of the ball ; a the radius of the ball, k the radius of gyration about C. Fii'st we will suppose the ball to be inelastic. For the motion of the ball after impact, we have mv^ = mu cos a - X (1), mv = Y - mu sin a (2), m^'w = Xa (3). Now, the ball being perfectly inelastic, the velocity of C at right angles to the vertical plane will be destroyed by the impact, or w = ; hence, from (2), Y = mu sin a (4). Also, the vertical plane being perfectly rough, the ball must roll without sliding after impact ; hence we must have aw = v^, and therefore from (1), (3), we get li~ (mu cos a - X) = Xa^, ._^ mk^u cos a ,.^ X = — a ,2 (5;. a + A; Next, let us suppose the ball to be elastic, e denoting the elas- ticity ; then v^, v ' , w, denoting on the new supposition what V , V , w, were taken to denote on the old one, we shall have mv^ - mu cos a - (1 + e) X (6), mv ' = mu sin a-(l+e)Y (7), ?«^V = (I + e) Xa (8), 444 impulsivp: forces. From the equations (5) and (6), we obtain (1 + e) k^u cos a a' - eU V = u cos a - \ ^r = —5 -^ u cos a ; a + (^ «- + k' from. (4) and (7), v^ = u sin a -.(1 + e) u sin a = - eu sin a ; and from (5), (8), , (1 + e) au cos a a + kr which values of v^, v ', to', completely determine the subsequent motion of the ball. (3) A perfectly rough sphere is placed upon a perfectly rough horizontal plane which is made to rotate with a uniform angular velocity about a vertical axis ; to determine the path described by the sphere in space. Let Oz (fig. 219) be the vertical axis about which the plane revolves ; let Ox, Oij, be any two horizontal lines fixed in space and at right angles to each other ; P the point of contact of the sphere with the revohdng plane at any time t; draw PM parallel to yO. Let OJ/=a;, PM=y; a = the radius of the sphere, m = its mass, ?)ik' = its moment of inertia about a diameter ; w = the angular velocity of the revolving plane about Oz, the motion being supposed to take place in the direction indicated by the arrow in the plane xOi/ in the figm-e; let X, Y, denote the re- solved parts of the friction exerted by the plane on the sphere, estimated parallel to Ox, Oy, respectively; J, w', the angular velocities of the sphere about diameters parallel to Ox, Oy, the directions of these velocities being estimated in the manner indicated by the arrows in the planes yOz, zOx. For the motion of the centre of gravity of the sphere we have "•§=^ (»' "•§=^ w^ and, for the rotation of the sphere, "''''4-^''' (^)' mk'^=-Xa (4). IMPULSIVE FORCES. 445 From (1) and (4) we have, eliminating X, "ir-^'^t (*>' and from (2), (3), eliminating Y, a ~:r^ = k —- (6). df dt - ^ Integrating the equations (5), (6), and adding arbitrary con- stants, we have a —- = C- ¥w (7), dt a^ = C' + Jej (8). dt ^ ' Now the linear velocity of the centre of gravity of the sphere relatively to the rough plane, in consequence of the rolling of the sphere, will be aw parallel to Ox, - ato' parallel to Oy ; and the linear velocity due to the rotation of the rough plane will be _ ^y parallel to Ox, wx parallel to Oy ; hence, -^ } -r-> being the whole linear velocity of the centre of the sphere parallel to Ox, Oy, respectively, we have dx „ dii , dt -^ dt and therefore, by the aid of (7) and (8), eliminating w" and w', dx dt ^ F/ dx\ f a^\dx aC '^ - 7 {"'■"' dt)' ['^eJai-W-""' dy ^, Ti' ( dy\ /, a^\dy aC a -f. = C +—[ OJX - -/], 1 + 7-J ^ = -7T- + f^^ > dt a\ dtj\ ky dt k^ eliminating t between these two equ.ations, we get iciC - k'wy) dy = (aC' + k^t^x) dx : integrating and adding an arbitrary constant C" , we obtain 2aCy - k'ojy" = 2aC'x + k^wx' + C" , „ , 2aC' 2aC C or X + y + -^ X - y:r- y + ji;- = (9). 446 IMPULSIVE FORCES. We proceed now to the determination of the arbitrary constants. Let the initial distance of the centre of the sphere from the axis of z be 5, and let the axis of x be so chosen as to pass through the initial position of the point of contact of the sphere M-ith the rough plane. Then, since the initial impulse of the friction of the revolving plane upon the sphere is at right angles to the axis of X, vre shall have initially — =0, w ' = ; hence from (7) we see (XT/ that C = 0. Again, F denoting the impulse of the friction when the sphere is just placed upon the revolving plane, and [ -y- clxi (w'), denoting the values of y- , w', just after the impulse, we shall have Kt)=^ ''''' mk\io') = Fa (11). But, since there is no sliding between the sphere and the plane, it is clear that 'chf ( ^^ , hio-a{J) (12), at where 5w is the velocity of the centre of the sphere parallel to Ox due to the rotation of the plane, and - a {J) the velocity estima- ted in the same direction due to the rolling of the sphere along the plane : hence from (10) we have m {hw - a (w')} = F, and therefore, by (1 1), a{hu)-a {J)] = k- {J), {J) = -^-^t. 5 and then, by (12), r7w\ ^ a^hta Ji^hui -^ = 5w - — — -, = - — - . at J a' + k a' + k But from (8) we have hence, putting for ( y- J and (w') their values. IMPULSIVE FORCES. 447 aoh (1) ^, cibk'it) ^, C + -^^-T, , C" = 0. Since (7=0 and C" = 0, we have, from (9), k'u) but X =^b when i/ = 0, and therefore A. W and we get for the equation to the path of the centre of the sphere in space ^^ ^. ^,2 _ p the equation to a circle having for its centre. (4) An inelastic homogeneous cylinder rolls down a perfectly- rough inclined plane which terminates in a perfectly rough horizontal plane ; to find the velocity of the cylinder along the horizontal plane, and the blow which it receives when it first impinges upon it. Let a = inclination of the inclined plane to the horizon, ?n = the mass of the cylinder, u = the velocity of its axis the instant before and u' the instant after impact, F = the initial impulse of the friction of the horizontal plane, estimated in the direction of the sphere's motion along it, and B the normal impulse of the horizontal plane ; then ^ = 5 (1 + 2 cos a) u, F= 2 ^'^^^ (1 ~ cos a), J5 = 9)IU sin a. (5) A homogeneous cylinder slides, without rolling, down an inclined plane which is for a certain space quite smooth, and after acquiring a given velocity is suddenly caused by the rough- ness of the surface to roll without sliding : to determine the velocity of the axis of the cylinder the instant rolling commences, and to find the initial impulse of friction. If u be the velocity of the cylinder the instant before and ^l' the instant after the commencement of perfect rolling, m the mass of the cylinder, F the initial impulse of friction ; then w = § w, F= I mu. APPENDIX. A COPY of Bernoulli's programme,* which had been received by the celebrated David Gregory, is at the present time in the possession of Mr. Gregory, Fellow of Trinity College. The following reprint of the challenge will probably be acceptable to those who take an interest in the antiquities of science. Acutissimis qui toto orhe jlorent Mathematicis. s. p. D. JOHANNES BERNOULLI, Math. P. P. " Cum compertum habeamus, vix quicquam esse quod magis excitet generosa ingenia ad molicndum quod conducit augendis scientiis, quam difhcilium pariter et utilium qusestionum propo- sitionem, quarum enodatione tanquam singulari si qua alia via ad nominis claritatem perveniant sibique apud posteritatem seterna extruant monumenta : Sic me nihil gratius Orbi Mathe- matico facturum speravi quam si imitando exemplum tantorum Virorum Mersenni, Pascalii, Fermatii, praesertim recentis iUius Anonymi \^nigmatistae Florentini aliorumque qui idem ante me fecerunt, pra:!stantissimis hujus eevi Analystis proponerem ahquod problema, quo quasi Lapide Lydio suas methodos ex- aminare, vires intendere et si quid invenirent nobiscum com- municare possent, ut quisque suas exinde proraeritas laudes a nobis publice id profitentibus consequeretur. " Factum autem illud est ante semestre in Actis Lips. m. Jun. pag. 269. Ubi tale problema proposui cujus utilitatem cum jucunditate conjunctam videbunt omnes qui cum successu ei se applicabunt. Sex mensium spatium a prima publicationis die Geometris concessum est, intra quod si nulla solutio prodiret in lucem, me meam exhibiturum promisi : Sed ecce elapsus est * See page 259. APPENDIX. 449 terminus et nihil solutionis comparuit ; nisi quod Celeb. Leib- nitius cle profundiore Geometria prseclare meritus me per literas certiorem fecerit, se jam feliciter dissolvisse nodum pulcherrimi hujus uti vocabat et inauditi antea problematis, insimulque humaniter rogavit, ut prsestitutum limitem ad proximum pascha extendi paterer, quo interea apud Gallos Italosque idem illud publicari posset nullusque adeo superesset locus ulli de angustia termini querelae ; Quam honestam petitionem non solum indulsi, sed ipse banc prorogationem promulgare decrevi, visurus num qui sint qui nobilem banc et arduam quaestionem aggressuri, post longum temporis intervallum tandem Enodationis compotes fierent. Illorum interim in gratiam ad quorum manus Acta Lipsiensia non perveniunt^ propositionem bic repeto. PROBLEMA MECHANICO-GEOMETRICUM DE LINEA CELERRIMI DESCENSUS. " Deierminare lineam curvam data duo pmicta in diversis ab horizonte distantiis et non in eadem recta verticali posita con- nectentem, super qua mobile propria gravitate decurrens et d superiori puneto moveri incipiens citissime descendat adpunctum inferius. " Sensus problematis bic est, ex infinitis lineis quae duo ilia data puncta conjungunt, vel ab uno ad alterum duel possunt eligatur ilia, juxta quam si inciurvetur lamina tubi canalisve formam habens, ut ipsi impositus globulus et libere dimissus iter suum ab uno puneto ad alterum emetiatur tempore brevissimo. " Ut vero omnem ambiguitatis ansam prsecaveamus, scire B.L. volumus, nos hie admittere Galilsei hypotbesin de cujus veritate seposita resistentia jam nemo est saniorum Geometrarum qui ambigat, Velocitates scilicet acquisitas gravium cadentium esse in suhduplicata ratione altitudinum emensarum, quanquam alias nostra solvendi methodus universaliter ad quamvis aliam hypotbesin sese extendat. " Cum itaque niliil obscuritatis supersit, obnixe rogamus omnes et singulos hujus sevi Geometras, accingant se promte, tentent, discu.tiant quicquid in extremo suarum methodorum recessu absconditum tenent; Rapiat qui potest prsemium quod Solutori paravimus, non quidem auri non argenti summam quo G G 450 APPENDIX. abjecta taiitum et niercenaria conducuntur ingenia, a quibus ut nihil laudabile sic nihil quod scientiis fructuosnni expectamus, sed cum virtus sibi ipsi sit merces pulcherrima, atque gloria immensum habeat calcar, ofFerimus prsemium quale convenit ingenui sanguinis Viro, consertum ex honore, laude et plausu, quibus niagui nostri Apollinis perspicacitatem publice et privatim, scriptis et dictis coronabimus, condecorabimus et cele- brabimus. " Quod si vero festuni paschatis prseterierit neniine deprehenso qui qufesitum nostrum solverit, nos quae ipsi invenimus publico non invidebimus ; Incomparabilis enim Leibnitius solutiones turn suam tum nostram ipsi jam pridem commissam protinus ut spero in lucem emittet, quas si Geometrse ex penitiori quodam fonte petitas perspexerint, nulli dubitamus quin angustos vulgaris Georaetriae limites agnoscant, nostraque proin inventa tanto pluris faciant, quanto pauciores eximiam nostram qusestionem soluturi extiterint etiam inter illos ipsos qui per singulares quas tantopere commendant methodos, interioris Geometriae latibula non solum intime penetrasse, sed etiam ejus pomceria Theorematis suis aureis, nemini ut putabant cognitis, ab aliis tamen jam longe prius editis mirum in moduni extendisse gloriantur. PROBLEM A ALTERUM PURE GEOMETRICUM, QUOD PRIORI SUR- NECTIMUS ET STREN.E LOCO ERUDITIS PROPONIMUS. " Ab Euclidis tempore vel Tyronibus notum est ; Ductam utcunque a puncto dato rectam lineam, a circuli peripheric ita secari ut rectangulum duorum segmentorum inter punctum datum et utramque peripherise partem interceptorum sit eidem constanti perpetuo sequale. Primus ego ostendi in eod. Actor. Jun. pag. 265. banc proprietatem infinitis aliis curvis convenire, illamque adeo circulo non esse essentialem. Arrepta hinc occasione, pro- posui Geometris determinandam curvam vel curvas, in quibus non rectangulum sed solidum sub uno et quadi-ato alterius segmentorum sequetur semper eidem ; sed a nemine hactenus solvendi modus prodiit; exhibebimus eum quandocunque de- siderabitur : Quoniam autem non nisi per curvas transcendentes qusesito satisfacimus, en aliud cujus solutio per mere algebricas in nostra est potestate. Quceyntttr Curta, ejus proprietatis , ut APPENDIX. 451 duo ilia segmenta ad qnamcunque potentiam datam elevata et simid sutnta faciant uhique unam eandemque summam. " Casum simplicissimiim existente sc. numero potentise 1, ibidem in actis pag. 266. jam solutum declimus, generalem vero solutionem quam etiamnum jDiemimus, Analystis eruendam relinquimus. Dabam Groningse ipsis Cal. Jan. 1697. Groningee, Typis CalharhitB Zandt, Provincialis Academic Typographic, ]G97. THK KNI). CAMBKIDGE; PRISTF.D BY METCALFE AND PALMER, Til INITY-SIRE ET . ERRATA. Page Line Error. Correction. 17, last line, m n 2 23, 18, four-fifths five-eighths 27, 9, (3.r - Im) (x + wf (-V - 2m) + 2m^ 3 3 (x + ?«)* — m"^ 50. 16, 17, OA O'A' 56, 7 from bottom, 1 /.. * t s"* 68, 22, AC AG 69, 6 from bottom, e 4> 73, 13 from bottom, a 'la 79, 12, (Pa\h toil (Pa\ iQb) 124, 17, Q S 148, 3 from bottom, m^ »'<•+! 150, 16, 1 +e-(l - e) cos « 1 — e + (1 + e) cos a 163, 3, (aa')h 2 (aa')i 272, 13, centrifugal fo: rce inertia An additional table of Errata, should it be found necessary, may be had at the Publisher's on the 1st of June, 1843. v_ ^ r rig 3(1 \" J"'^ M I'i-9 ■ -fA McUai/f i Palmer, lidwc V ^-• ^'4-^ JtftecBLi/e APiUmer. LiatM>4 .r\ MeccaZ/e * falrt^r. Zuft^f Pv 7^ T^ 7i. ' MucaUe k PiUm-^r Liefto^ M^KtcaJte kTaZm^r.LtChoQ ^ rtf re fty <*■/. fifde Fig 77 A ■^. T r ru, ex T\ )l^ T^ Tig m. -^^^ Mtt4:al/e * Palmar. LtOiaq Fi^ t04 ^^ -"1^ Tig ■ //«. ■P^^ y' \ r ?<^ \ F Tiy. IM. Fty. 433. ?w /J/ ^'S ''V. ^'/■yj^. T'y /^J, > r^ u^ F,^./46 r,^ /^7 ft-f. ua Fly. M». r,y. /so Fiy /Jty Jf,/r«i/r * PtImtr.Ltl Tig. 160 •^ — "jr, Fiy /*■/ Fiy 174 ,0 / f'y ns Mei£a2h k P^