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THEORY OF COUPLES. 



THEORY OF COUPLES. 



210 



the orbits of binary stars: they would call hiui a practical man. I 

 should give him quite another name if he took up steam or star for 

 anything beyond relaxation, supposing his taste to turn that way. 

 The disposition to hold material application to be always practical is 

 one of the consequences of the want of psychological thought, and will 

 vanish before sound logical training, with other myopisms." (' Cambr. 

 Phil. Trans.,' vol. x. p. 1.) 



In conclusion, the word practice as opposed to theory takes an 

 advantage from its meaning as opposed to profusion. There are many 

 persons who have so hazy a view of the two meanings as to imagine 

 that the two antitheses are one and the same. 



THEORY OF COUPLES. The two motions of which any rigid 

 system is susceptible are those of TRANSLATION and of ROTATION. 

 Each of these has this peculiarity, namely, that one particular case of 

 its application yields the other kind of motion. Every motion of a 

 system can, for any one instant, be resolved, at most, into a motion of 

 translation of the whole system, combined with a motion of rotation 

 about an axis ; and every application o a system of forces to any rigid 

 body, produces, generally speaking, this compound of translation and 

 rotation. Also, if equal and opposite forces, such as would produce 

 simple translation, be applied at the same point, or if equal and oppo- 

 site forces, such as would produce rotation, be applied about the same 

 axis, the result is that the equilibrium, or previous motion, of the 

 system remains undisturbed. 



But if the equal and opposite forces of translation be applied at 

 different points, the result is rotation only, for the first instant ; and 

 if the equal and opposite forces of rotation be applied about axes not 

 coinciding, but only parallel, the effect, at the first instant, is transla- 

 tion only. And though the doctrine of motion is now properly ex- 

 cluded from statics, yet the preceding theorems, together with others 

 mentioned in ROTATION, should be well understood, and viewed in 

 connection with the science of equilibrium, which is always illustrated, 

 though it may not be demonstrated, by such considerations. 



It was for a long time a curious but barren exception, that though 

 any two forces acting in the same plane may, generally speaking, have 

 their joint effect supplied by one single third force, yet if the two 

 forces be equal in magnitude, and opposite in direction, no such single 

 third force will do. If indeed they be applied in the game line, as o r 

 and Q u in the first figure, they equilibrate each other ; but if not in 



Q R 





tho same line, as o p and Q R in the second figure, no one single force 

 can be found which will either equilibrate them, or produce their 

 effect. Some years ago, M. Poinsot, already mentioned for his 

 beautiful theory of ROTATION, applied a remarkable theorem con- 

 nected with such pairs of forces to the establishment of the theory of 

 the statics of rigid bodies, in a manner which has made his system 

 rapidly take its place among the fundamental bases of the science. We 

 shall in this article point out the manner in which this can be done, 

 without much demonstration, with a view to draw the attention of 

 those who have learned the doctrine of equilibrium in the old way : 

 we cannot make it intelligible (without too great length) except to 

 those who have learned the principles of analytical statics. 



M. Poinsot called a pair of equal and opposite forces, not equili- 

 brating each other, by the name of a couple too general a term 

 perhaps : by it is to be understood a couple which cannot be made 

 anything but a couple, or cannot be replaced by one force : an incom- 

 couple. The plane of the couple ia the plane drawn through 

 the parallel forces ; the arm of the couple is any line drawn perpen- 

 dicular to the f urces from the direction of one to that of the other ; 

 the a.rli of the couple is any straight line perpendicular to its plane. 

 And if we consider any axis, it will be apparent that the moment or 

 leverage of the couple [LEVER] to turn the system about that axis is 

 represented by the product of one of the forces and the arm. For if, 

 with reference to the axis, x be the arm of one of the forces, x J; is 

 that of the other, a being the arm of the couple. Hence if p be one 

 of the forces, the united leverage is P (x + o) Fx or J; PO. This 

 product P a is called the moment of the couple. 



The last-mentioned property will give a high probability of itself to 

 the following theorems, which are the basis of the theory of couples, 

 and can be proved, the first by aid of the composition of forces only, 

 the second by the principle of the lever. Any couple may have the 

 direction of its arm changed, and consequently of its forces, in any 

 manner whatsoever, either in ite own plane, or in any plane parallel to 

 it, provided only that the direction in which it tends to turn the sys- 

 tem remains unaltered. Secondly, any couple may be replaced by 

 another which has the same moment, the plane and direction of 

 turning remaining unaltered ; that is, the arm may be shortened or 

 lengthened in any manner, provided the forces be increased or dimi- 

 nished hi the same proportion. If the system were in equilibrium 

 before, it will remain in equilibrium, however its couples may be altered, 

 in any manner described in the above theorems. Hence it follows 

 that a couple is entirely given when there are given : 1. Its axis or 

 any line perpendicular to its plane, which is also perpendicular to any 



ART3 AND SCI. DIV. VOL. VIII. 



of the planes into which it may be removed. 2. The moment of the 

 couple ; specific forces or arms are unnecessary for its description, so 

 long as their product is given. 3. The direction in which it tends to 

 turn the system. The easiest way of describing a couple is then as 

 follows ; suppose, for example, a horizontal one : Take any vertical 

 line for the axis of the couple ; on that axis lay down a line propor- 

 tional to its moment, and agree that vertical lines drawn upwards shall 

 represent moments tending to turn the system from west to east ; and 

 downwards, those tending to turn the system from east to west. But 

 a sign must also be agreed upon ; positive moment must consist in 

 tendency to turn in one direction, and negative in the other. 



The composition and resolution of couples is easily shown to be 

 done in a manner which perfectly resembles that of ROTATIONS. When 

 the couples can have a common axis (act in the same plane or parallel 

 planes), the moment of the resultant is, in sign and magnitude, the 

 sum of the momenta of the components, with their proper signs. To 

 find the resultant of two couples which cannot have a common axis, 

 take axes to them which pass through the same point, and on these 

 axes lay down lines representing the moments of the couples in their 

 proper direction. On those lines complete a parallelogram ; the 

 direction of the diagonal is the axis of the resulting couple, and its 

 length represents the moment of that couple. Care must be taken to 

 lay down the directions of the moments properly on the axes ; the best 

 isolated rule (when reference is not made to distinct co-ordinate planes) 

 is as follows : Let the parts of the plane of the axes which lie in the 

 angle made by the lines representing moments be turned by the two 

 couples in opposite directions. To the student to whom such a 

 direction would be useful, we should say, appeal in all cases to the 

 perceptions derived from ROTATION. 



To apply the preceding theorems to the statics of a rigid body, we 

 first take the following conventions : Assume an origin and three 

 rectangular axes of co-ordinates, as usual. Let the forces which act at 

 each point of the system be decomposed into three, parallel to the 

 axes of x, y, and -. Let each force be called positive, when it acts 

 towards the positive part of the axis to .which it is parallel ; if, for 

 instance, the axis of z be vertical, and if its positive part tend upwards, 

 all forces in the direction of z, wherever they act, are called positive 

 while they act upwards, and negative when downwards. As to couples, 

 let their moments be called positive when, acting in the planes of x 

 and y, y and z, z and x, they tend to turn the positive part of the first- 

 named towards the positive part of the second (jcy, yz, zx). Let P, bo 

 the first point of the system ; let its co-ordinates be .r,, y l( z l ; let the 

 forces in the three directions acting at that point be X 1( Y,, z,. Let r a 

 be the second point ; x s , y a z s , its co ordinates ; X 2 , Y 2 , z a , the forces 

 there applied; and so on. All co-ordinates and forces have their 

 proper signs. At the origin apply the following pairs of equilibrating 

 forces : x, and x,, Y t and YJ, x t and .z t ; X 2 and x.,, Y 2 and Y 2 , 

 x, and Z 2 , and so on ; which of course do not affect the equilibrium, 

 and are over and above those already applied. Again, at the extremity 

 of .r,, in the axis of .r, apply the equilibrating forces Y 1( Y, ; at tho 

 extremity of y,, in the axis of y, apply z,, z, ; at the extremity of z,, 

 in the axis of z, apply x,, x, ; and so on for the other points, Lastly, 

 let the points of application of the original forces x,, Y,, z,, be changed 

 so that each shall act at the projection of the point of application made 

 by its co-ordinate ; and the same for the other points. Nothing is 

 done but the application of mutually destroying forces, or the change 

 of the point of application of a force to another point in its direction, 

 and the following figure will show the present arrangement for one 

 point. The original forces, transferred, are marked x, Y, z ; the original 

 point of application, r ; and the other forces, equilibrating two and two, 

 have great and small letters at their extremities. 





We now see that the forces x, y, z, are equivalent to 



1. The forces x, Y, z (marked A, B, c) applied at the origin. 



2. A pair of couples to the axis of z (L, 4) (x, n), the first positive 

 with the moment tx, the second negative with the moment xy. These 

 two are equivalent to one couple with the moment \'x xy. 



r 



