THE 



338 



THE 



not 



theiimv t,MO 



K> well it- 



theory, against practice, as pn' 



Utter nttoaswu 



fchare of t: 



the most recent practical knowledge -o the other 



in- ontii of W] 

 fret part of the mass of <! 



that part which is not yet off the anvil. Suppose n mer- 

 chant cuing into the bail court to prove 



her hi I 



ould pmi 1 sum: 



,d n- 



ssful. if it would not ho so. over and over again. 

 he is further qr positively 



icss will make you worth the sum in 

 on.' "' 1 cannot.' ho rc;ilie<. swear any 



such thin:; ; but I have enough not employed in business, 

 in land and mortgages, and in the funds! to pay twenty 

 shillings in tho pound five times over, upon every risk 

 which I sm liable to. 1 What would be thought of counsel 

 who should retort, ' That is nothing to ug ; you are described 

 as a merchant, and your solvency must be tried by tho 

 state of that part of your property which is now under- 

 going the fluctuations of trade?" S:u-h is and always 

 be the state of theory ; the amount which is actually 

 realised is enormously greater than the floating balance 

 which is being: worked out. Those who arc encased in 

 producing: fixed capital from the latter, have a risrht to the 

 credit which arises Scorn the interest of the former: their 

 labours for the time being are not to produce their return 

 at the instant. 



We have, in compliance with common notions, not ad- 

 verted to the consequences of theory upon the mind and 

 thoiichts of men, but have treated it as if its sole 

 were to advance the mechanical arts and better the phy- 

 sical condition of society. But this is under protest that 



if it conld not be proved that rational ii 

 of nature had added one single atom to 1 com- 



fort of life, there would remain such an enormous K 



I ameliorations which can be traced to that source as 

 would outwejeh even the triumphs of steam. 



THEORY OF COUPLES. The two motions of which 

 any rigid system is susceptible arc those m ' Ti: \N STATION 

 and of ROTATION. Each of these has (hi- 

 namely, that one particular case of its application yields 

 the other kind of motion. Every motion of a system can, 

 for any one instant, lie resolved, at most, into" a motion 

 of translation <v, :n, combined with a motion 



of rotation about an ] even- application tit 



tern of forces to any ri^id body, produ- -. 

 impound ion and r- 



equal and opp 



translation, be applied at tho same point, or if equal and 

 opposite forces, such as would produce rotation, be ap- 

 plied about the same a\i ,lt is that the equili- 

 brium, or previous motion, of the system remains undis- 

 turbed. 



But if the equal and opposite forces of tr.i' 

 applied at different points, the result is rotation only, for 

 the first instant : and if the equal and opposite for 

 rotation be applied about axes not coinciding, but only 

 parallel, t) at the first i only. 



And o doctrine of motion is now properly ex- 



cluded from si ceding- theorems, to 



with others mentioned in H 

 stood, and viewed in conr 

 librium, which is always illustrated, thuu 

 demonstrated, by si ns. 



It was for a long time a o 

 though any tw<- 

 rally speaking, have their joint < 



third force, yet if the two f<" itwlc, 



and opposite in direction, no such single third force will do. 

 If indeed they he applied in the same line. :is OPandQ R 

 in the first figure, they cquililvatr each other: but if not 

 in the same line, as O P and Q R in the seeom 



them, or produe, 



M. 1'omsot, aln- 







one single fore- < mj which will either equilibrate 



f forces to the establishment of the theory 



:ade 

 his system rapidly take its place 



shall in this article point out 



the manner in which this can be done, vir 'i do- 



to draw the attention of 



who have learned the doctrine of equilil rium in the old 

 AC cannot make it intel'i 



to those who ha'. ' the piin. 



M. Poin&ot called a pair of equal and opposite fn: 

 not equilibrating each other, by the nam 

 too general a term p, it is 1u be understood a 



couple which cannot be made anyt! 

 cannot be replaced by one force : an ; 

 The }>/,irn' ot the couple is the plane drawn 11 

 parallel forces : the arm of the couple is any line drawn 

 perpendicular to the forces from the din 

 that of the other: the art's of the couple is a' 

 line perpendicular to its plane. And if \ 

 axis, it will be apparent that the moment or li- 

 the couple [T.KVxi;] to turn the system about 11 



l'\ the product of one of the forces and the 

 arm. to the a\is. .r be ' 



one of the forces, .r^-a is that of tl 

 arm of the couple. Hence if P one of the t 

 united leverage is P (.rn) Pr or Pa. This pro- 

 duct Pa is called the mumrnt of t 

 The last-mentioned property will (rive a high probni 



'f to the following theorems, which 



the theory of couples, and can be proved, the first by aid 

 of the composition offerees only, the second by the prin- 

 ciple of the lever. Any couple may have the direction of 

 its arm ch.v.iircd. and consequently of its forces, i:; 



, or, either in its own plane, or in any piano 

 1 to it. provided only that the direction in which it 

 to turn t!. remains unaltered. Secondly, 



any couple may be replaced by another which has the 

 same moment, the plane and direction of turninjr remain- 

 ing unaltered; that is, the arm nir; 



nod in any manner, provided the forces be in- 

 I or diminished in the same proportion. If thi 

 tern were in equilibrium before, it will remain in equili- 

 brium, however its couples may be altered, in any man- 

 ner described in the abo\c theorems. Hence it follows 

 that a couple is entirely i;iven when there are given : 1, 

 Its axis or any line perpendicular to its plane, which is also 

 perpendicular to any of the planes into which it may be 

 removed. 2, The moment of the couple : specific I 

 or arms arc unnecessary for its description, so Ion;: as their 

 product is given. :$. The direction ih which r 

 turn the system. Tin- : a couple 



is then as follows : suppose for example n horizontal one : 

 Take any vertical line for the axis of the couple, on that 

 ay down a Hue proportional to its moment, and 

 \ertical lines drawn upwards shall represent 

 monic' : to turn ' 



and downwards. ' nj to turn the system from 



L'n must also be agreed upon ; posi 1 

 moment m' m tendency to turn in one direction. 



;ati\o in the other. 

 The composition and resolution of couples 



:i manner which peru 

 . Wlu-n tlie couples can have a common 

 the same plane or parallel plain 

 .!ant is, in si<;n and magnitude, the sum of the 

 c-omponente, with their pri To 



find the resultant of two couples whu ' ' nave 



take axes to them whiei 

 . and on these axes lay down li 



i their proper 



On those lim -,- a parallelogram : 



resulting 

 ' 



,:ist be taken to la\ 

 properly 01 



M the par! Njeofthe : 



which lie in the an-le made by the lines representing mo- 

 ments be Umie.l I)) tin .'Its in opposite directions. 



