1M 



iltl'M. 



KOTATIOX. 



i-a 



it'... - ' .. 



Smith. 



TO-SI; 



4-814 



-J4 



100-000 



Mailer. 



JJ-J 



9-89 

 18-li 



I, ,! I 



At present the data are innuflicifiit fur the calculation of it- formula. 



KOSTKUM. or, more properly, KO.STRA, wo a platform or elevated 

 sfMoeof grounti in the Koman fonim, from which the orators u--d t 

 addm the people, and which derived iU name from the circumstance 

 that altar the conquest of l-atii.m the beaks (rostra) of the Antiatitui 

 hip. w* affixed to the front of it. (l.iv.. viii. 14.) The rostra W:H 

 lulman the Comitium, or place of auembly for the Ciiriio, ami the 

 Porom, properly to called, or place of assembly for the Couiitia Tributa. 

 Pnnssii. in hi* work on the Roman Kormn, >iuoted by Arnold 

 ( HMory of Rome,' TO!, it, p. 16fi>, judging from the view* of the 

 runtra given on coins, supposes that " It wu a circular building, mixed 

 on arcoM, with a stand or platform on the top, bordered by a jsurapet, 

 the JI,M to it being by two flight* of steps, one on each side. It 

 fronted towards the comitium, and the rostra were affixed to the front 

 of it, just under the arches. Its form has been, in all the main point*, 

 preserved in the ambones, or circular pulpits, of the most ancient 

 ^fan^Jwi., w hkh also had two flights of steps leading up to them, one 

 ..... . . .. ;. by which the preacher a* , n.i. ,1. ,,nd another UM 



west aide, for his descent. Specimens of these old pulpits are still to 

 be seen at Rome, in the churches of S. Clement and S. Loren. 

 le Mure." The orators appear to have walked up and down the rostra 

 in addressing the people, and did not, like modern speakers, remain 

 standing in one spot. Down to the time of Cuius Gracchus even the 

 tribunes in speaking used to front the comitium ; but he turned bin 

 back to it, and spoke with his face towards the forum. (Niebuhr, 

 'History of Rome,' vol. i.. note 990; vol. iii., note 268.) 



KI "TATION (Rota, a wheel). The popular conception of a body in 

 rotation is vague, except only in the cose in which the rotation is made 

 about an immoveable axis. This subject has accordingly been usually 

 treated by mathematical methods ; and mathematicians content with 

 their result*, and with their power of interpreting them, did nothing 

 towards the improvement of the manner of presenting the elementary 

 Tiw of filiation. M. Poinnot first divested the subject of its previous 

 complexity, in a Memoir read to the Academy of Sciences, May 19, 

 1834. 



There is thi.i parallel between the conception we form of the simple 

 motion of a point and that of a solid body, namely, that each has a case 

 of peculiar simplicity, by which others are rendered more easy to 

 describe. A point may move in a straight line, or may preserve its 

 direction unaltered ; a body may revolve round a fixed axis, or each 

 point may preserve iU circle of revolution unaltered. But owing to the 

 comparative simplicity of the motion of a point, it is easy [DIRECTION] 

 to carry with us, when it moves in a curve, the idea of its still having 

 a different direction at every point of the motion, namely, that of the 

 T of the curve. It is not so easy to see that whenever a body 

 mores about a fixed point, no matter how irregularly, there is always, 

 at every instant of the motion, some one axis which is, for that instant, 

 at ret. Thin notion of an instantaneous axis of repose, not continuing 

 to be such for any finite time answering to that of au instantaneous 

 direction in curvilinear motion, which does not continue for any linitu 

 time to represent the direction must be first distinctly formed, before 

 any satisfactory account of the rotation of a body can be given. 



Let us suppose a uniform sphere, with a fixed centre, but otherwise 

 free to move in any way. Let a succession of forces act upn it. 

 gradual or not, in such a manner that it will never move round one 

 axis for any finite time during the continuance of their action. At a 

 certain moment, let all the forces cease entirely, leaving the sphere to 

 itaelf. It is easy enough to see that from and after the moment of 

 discontinuance, the sphere will move round an axis which remains 

 unaltered. There must then, at the very moment of discontinuance 

 of the forces, have been an axis which was for that moment at rest, 

 namely, the axis on which the motion is to continue after the forces 

 cease. In this way, know-in/ that curvilinear motion would become 

 rectilinear the moment that the deflecting forces are removed, we may 

 form an idea of the tangent of a curve, the line of direction for the 

 time being. 



One of M. Poinsot's remarkable propositions is the following : Any 

 motion of a system round a fixed point may be attained by cutting a 

 cone (in the roost general sense of the word) out of the body, with 

 the fixed point for a vertex, and fixing in space another cone for it to 

 roll upon, also with the fixed point for a vertex. 



Thus if, in the adjoining diagram, the cone A be made to roll 

 upon the cone B, both being supposed destitute of impenetrability, so 

 that the contact of the curves of A and B can always be mode, ami if 

 the system out of which A is cut be then restored (also without im- 

 penetrability), there will be a complete geometrical representation of 

 ible motion of the system about c. Moreover, there is no 

 motion which might not be represented in the same manner 

 iperly choosing the cones A and B, and the axis of repose for the 

 instant is the line in which the two cones touch. 



If we suppose no fixed point in the system, so that motion of trans- 

 lation, M well as of rotation, is possible, M. Foiusot has given another 



equally distinct notion of the state of the motion during an infinitely 

 small time. The most simple notion which we can form of a com- 



bined translation and rotation is the screw-like motion, in which a 

 uniform motion of translation is accompanied by a uniform m K 

 rotation round a linn parallel to the motion of translation. M. Poinsot 

 has shown that every motion of a system must be, at any one. instant, 

 either a simple motion of translation, or one of rotation, or the screw- 

 like motion above described. That is to say, at every point of time in 

 the motion of a system there exists a line (whether internal or e.\i . i n.il 

 to the material system matters not, so long as they are itnmoveably 

 connected) along which the system id at that instant sliding, while all 

 the rest of the motion at that instant is simple rotation about that 

 tlijtphtg <i r/.-t. 



Let us now suppose a system to receive at the same time two 

 motions, round two different axes of repose : that is to say, given two 

 different motions, required the motion which will result from the two 

 motions impressed on the system at once. There will be at the first 

 instant an instantaneous axis of repose, which it is required to find. 

 First let the two axes pass through the same point A (Fiy. 1), and 



Fig. l. 



choose the angle B A c out of the four angles made by the two axes, 

 in such manner that points of the system lying in the .ingle B A c 

 would be elevated by the rotation round B A, and depressed by that 

 round C A, or rice rend. On the axes take AH and AC, lin.-s propor- 

 tional to the angular velocities about those axes, complete the paral- 

 lelogram AD, and draw the diagonal AD. Then AD i- 

 repose at starting (which however it may not continue to be), and A i> 

 represents the angular velocity round that axis at starting, in the sum' 

 manner as A B and A c represent the impressed angular velocities about 

 A Band AC. [COMPOSITION.] 



Next let the axes be parallel to one another, say perpendicular to the 

 plane of the paper, passing through A and u (Fly. -2). If the rotations 



be such that A and n would both rise, or both fall, on the piper, 

 each by the rotation about the other, take a point c in B A produced. 

 nearest to the axis about which the angular velocity is greatest (say 

 that of A), and such that c A is to c B as the angular velocity about 11 

 to the angular velocity about A.. Then the axis of repose at starting 

 is a line passing through c parallel to the 'former axes, and the angular 

 velocity is the difference of the angular velocities about A and n. and in 

 the direction of the greater. In this case the directions of the rotations 

 about A and B [DIRECTION OK MOTION] are different. There is one 

 remarkable cose, namely, when the rotations about A and B are equal. 

 In this case the rule would lead us to a rotation equal to nothim; 

 made about a point at an infinite distance one of those extreme con- 

 clusions which require interpretation. The fact is that these two 

 rotations give only a simple motion of translation = A B x Angular 

 Velocity per second, and such as to make the system move upwards or 

 downwards on the paper according as the separate rotations would 

 make the point* A and B move upwards or downwards. This parti- 

 cular case will be more intelligible when looked at with the help of 

 the TIIEORY OF COUPLKS. 



But if the rotations be in the same direction, so that A will be 

 lowered and B raised, or rice rend, each by the rotation about the 

 other : Take a point D, dividing A B so that AD is V r> " as the 

 angular velocity about B is to that about A. Then will the axis of 

 repose at starting be a parallel drawn through D to the a 

 ilin.uirh A and B, and the angular velocity will be the sum of the 

 angular velocities about A and u, its direction being that which lower.-, 

 A on the paper and raises B, or vice rend, according as is done by the 

 given angular velocities. 



Lastly, let the axes be neither parallel nor intersecting (1'iij. 3), as 

 ABandcD: Through the point m. in which CD meats the common 

 parpandieular, inn, draw r.v parallel to AB, and at the instant at 

 which the rotations round AB and CD commence, impress t\\. 

 and contrary rotations about E F, each equal to that about A B. These 

 produce no effect, so that the composition of the four rotations gives 

 the game result as that of the two. Now, as above stated, the rotation 



