surface or solid whose base is the line or figure given, and height equal to the arch described by the centre of gravity. (Fig. 1. Pl. VIII.) Let BDdb be the figure generated. On the base BCD erect the surface or solid BDFE, and let C be the centre of gravity. Since the arches Bb, Cc, Dd, are as the radii, AB, AC, AD, that is, as BG, CI, DH: therefore, if CI = Cc, then will all the lines BG, CI, DH, &c. all the arches Bb, Cc, Dd, &c.; that is, the surface or solid BDdb = BDHG, that is (by this Prop.) BDFE. Cor. 2. Also, if a curve revolves about any right line drawn through its centre of gravity; the surfaces generated (either by a partial or total revolution) on opposite sides of the line, will be equal." For (by Cor. 2. Prop. XLIV.) each part of the curve multiplied by the distance of its centre of gravity from this line, must be equal on both sides. And, by Cor. 2., each surface generated, is equal to the curve multiplied by the arch described at that distance; and these arches (being similar) are as these distances. Whence, each surface is as the curve multiplied by the distance of its centre of gravity; and, therefore, they are equal. PROP. LVI. (Fig. 2. Pl. VIII.) LET THERE BE ANY SYSTEM OF BODIES A, B, C, CONSIDERED WITHOUT WEIGHT, AND MOVEABLE ABOUT AN AXIS PASSING THROUGH S; AND IF ANY FORCE CAN GENERATE THE ABSOLUTE MOTION m IN A GIVEN TIME; IF THE SAME FORCE ACT AT P, PERPENDICULAR TO PS; THE MOTION GENERATED IN THE SYSTEM, IN THE SAME TIME, REVOLVING ABOUT THE AXIS AT S, WILL BE A X SA + B x SB + C x SC X SP x mi. A X SA'+ B x SB2+ C x SC2 For, suppose PS perpendicular to the axis at S, and to the line of direction PQ. And SA, SB, SC, perpendicular to the axis at S. And, suppose the force ƒ divided into the parts p, q, r, acting separately at P, to move A, B, C. Then (by Cor. 3. Prop. XIX.) the bodies A, B, C, will be acted on respectively with the forces SP SP SP SA P 9 SB SC Since the angular motion of the whole system is the same, the velocities of A, B, C, are as SA, SB, SC; and their motions as A X SA, B x SB, C x SC; and these motions are as their geSP SP SP Whence p, 1, r, are as nerating forces. 9, 7. A X SA' SP C x SC2 B x SB2 - put the sum of theses, and since p + ➡ force acting at A. Then f: m :: ƒ x A x SA mx A x SA motion of A. After the same : mx B x SB m x C x SC S are the motions of B and C. Therefore, the whole motion generated in the system is, AX SA + B x SB + Cx SC S Cor. 1. If you make SO X m. „A × SA' + B x SB2 + Cx SC' A X SA + B x SB + C × SC, then if all the bodies be placed in O, the motion generated in the system will be the same as before, as to the quantity of motion, or the sum of all the absolute motions; but the angular velocity will be different. cases, will be ×SP xm, and A+B+C×SO For the motion generated in these two A X SA + B x SB + C X SC A × SA' + B x SB2+ C × SC2 A+B+C XSO2 X SP x m; and if these be supposed to be equal, there comes A X SA' + B x SB + C x SC2 out SO = A X SA + B x SB + C × SC Cor. 2. The angular velocity of any system A, B, C, generated in a given time, by any force f, acting at P, perpendicular to PS, is as SP x f AX SA + B x SB2 + C × SC2 For the angular velocity of the whole system is the same as of one of the bodies A. But the absolute motion of A is = m × A + SA the angular velocity is as the absolute velocity directly, and the radius or distance reciprocally; therefore the angular vel. of A, and, Cor. 3. Hence, there will be the same angular velocity generated in the system, and with the same force, as there would be in a single body placed at P, and whose quantity of matter is A XSA + B x SB2 + C × SC2 For let P SP2 that body, then (by Cor. 2.) since ƒ and SP are given, the angular velocities of the system and body P, will be to one another, as 1 to. 1 AXSA+BX SB2, &c. PX SP2 Which be ing supposed equal, we shall have PA× SA2+B× SB3+C× SC2 SP2 Cor. 4. The angular motion of any system, generated by a uniform force, will be a motion uniformly accelerated. PROP. LVII. (Fig. 3. Pl. VIII.) TO FIND THE CENTRE OF PERCUSSION OF A SYSTEM OF BODIES, OR THE POINT, WHICH STRIKING AN IMMOVEABLE OBJECT, THE SYSTEM SHALL INCLINE 10 NEITHER SIDE, BUT REST AS IT WERE IN EQUILIBRIO. Through the centre of gravity G of the system, draw a plane perpendicular to the axis of motion in S. And if the bodies are not all situated in that plane, draw lines perpendicular to it from the bodies, and let A, B, C, be the places of these bodies in the plane. Draw SGO, and let O be the centre of percussion. Draw Af, Bg, Ch, perpendicular to SO, and Aad to SA, and make ad =SA, and draw ea perpendicular, and de parallel to SO. Then aA will be the direction of A's motion, as it revolves about S. And the system being stopt at O, the body A will urge the point a forward, with a force proportional to its matter and velocity; that is, as Ax SA or A x ad. And the force wherewith A acts at a in direction ea, is AX ea or A x Sf. And the force of A to Turn the system about O, is A x Sf x a0 (by Cor. 3. Prop. XIX.) = A × Sƒ × SO Sa Ax Sf x SO- A x SA'. Lakewise, the forces of B and C to turn the system about O, is as BX Sg x SO-B× SB, and C × Sh × SO-Cx SC'. And since the forces on the contrary sides of O destroy one another, therefore A × Sfx SO-AX SA2+B × Sg × SO-B× SB2+C× Sh × SO-C AX SA2+B × SB2 + C × SC2, &c. XSC0. Therefore SO= AXS/+BxSg+CxSh, &c. the distance of the centre of percussion, from the axis of motion. Where note, if any points f, g, h, fall on the contrary side of S, the correspondent rectangles must be negative, — A × Sƒ,— Bx Sg, &c. Cor. 1. IfG be the centre of gravity of a system of bodies A, B, C, the distance of the centre of percussion from the axis of motion, A X SA2 + B x SB2 + C x SC2, &c. that is, SO SG XA + B + C, &c. For, by Prop. XLIV., A × Sƒ + B x Sg + C × Sh= A+B+C × SG. Cor. 2. The distance of the centre of percussion from the centre GA' x A+ GB3 × B + GC2 × C of gravity G, is GO = SGXA+B+C For, AX SA2+BX SB2+C× SC2➡A × SG2+GA2— 2SG × Gƒ +BXSG+GB2+ 2SG × Gg +C × SG2+CG2+2SG× Gh, by Eucl. II. 12 and 13. But, (by Cor. 2. Prop. XLIV.) AX Gf+Bx Gg+C× Gh=0; therefore A× SA'+BxSB2+Cx SC2 A+B+C × SG2 + A × GA2 + B× GB + C × GC". SG x A+B+C Whence (by Cor. 1.) SO or SG + GO= For each of the bodies A, B, C, and their distances from G, are given. Cor. 4. Hence, also, if SG be given, GO will be given also. And, therefore, if the plane of the motion remain the same, in respect to the bodies, and the distance SG remain the same, the distance of O from G will remain the same also. Cor. 5. The percussion or quantity of the stroke at O, by the motion of the system, is the same as it would be at G; supposing all the bodies placed in G, and the angular velocity the same. For the sum of the motions of A, B, C, in the system, acting against O, is as A× SA × SB SC_AX SA'+BxSB'+C× SC2 +BX SBX +CXSC x SO SO SO SA SO =SGXA+B+C (by this Prop.) but SGX A+B+C, denotes the |