MECHANIC \L PHILOSOPHY. BTATIOR 



IX COUrtES. 



In a tiiiu 



Q' E H Q' will represent in mag- 



nitude and direction the couple Q'E'H, whioh will re- 

 CD 

 place Q -D C provided Q' H or Q'=- Q' EH < 



Completing the pa- Fir. i-(J). 



r,::-: pM*? IK> I! H 



on the plane of tho 

 till! r. and joining II R, 

 I H R its diagonal will 

 represent a force R 

 in magnitude and di- 

 rection, which will 

 replace the forces P' 

 and Q'. 



Since the plane of 

 the table is at rilit 

 an el en to the planes 



H '<>, 



and oomiequentlyKuc. , "* 



IV M . Prop. XIX., to their intersection EH. There- 

 fore H R in the plane of the table will be at right angles, v 

 t.. K II. V 



Similarly, P' and Q' acting at E will be replaced by a 



force R, represent, d ),y 1 : R acting at right angles to E H. 



M.-nce, on the whole, we have replaced the couples 



P A P. :n,d <,< D, by a single couple R- E H, whose arm 



-ection of the planes in which the couples 



act. 



Let us now suppose the parallelogram H Q R P , 

 which we have previously drawn on the plane of the 



be drawn as in (2), on the plane of the paper. 

 Draw H M perpriidi.-ul:ir t.. H Q', II O perpendicular 

 to H R, and H N perpsodMolar t.. H P'. 



Then if - angle QEP*. MHH will - 0, the angle 

 M n i. U ',' II i:. "'.d MIO - P'HR. 



Take H M - '.' 1 1 K. H <> - R'HE, and H N - 

 P 11 MO and NO. 



Then MM. M <>, and II N will be the axes of the 

 couple. Q'. H i:. i: M K. .-md P'-ll K. 



BecatiM H 1' I: ',' n a parallclojjram ; therefore, the 

 angle H 1" R 180'' and l>y Trigonometry, 



4- I" R 2 M I" I" R cos. H F R - 

 M P PJ + I"U-r- - M I" P R-cosfl 

 p 4. Q ' + 2 P- Q cos. 



;. lying Iwth iiide* of the above equation by H E*, 

 we have 



- P' M E + Q H E 1 + 2 P-H E Q- 

 i H E cm. 9 



UM-Q-HE.HO-R HE, andHN-P HE 



Heooe, iuUtitnting these values in the equation, wo 



This is an equation identically tho name as that we 

 ild arrive at if we supi).r II M ON a 

 parallelogram whose diagonal U H (). 



Hence, if two aides of a parallelogram 

 represent the axes of two coinpoi,. i.t n 

 its diagonal represents the axes of tho resul- 

 tant couple. 



Similarly it may be shown that if tho throe 

 edges of a parallelepiped represent the axel 

 of three component couples, tin diagonal of 

 the parallelepiped will give the magnitudo 

 and direction of the axis of the resultant 

 couple. 



PROPOSITION XVL 



When any number of couplet aft in the 

 tame, or in jxiru/M p/<ie,i ; themi. 

 the retultant couple it the algebraical turn 

 of the momenti of the component couplet. 



Let P, -A, B,, P. -A, B,, P, -A 3 B s be 

 three positive, and P 4 -A. t B 4 a negative 

 couple acting in the same plane (Fig- 55 



Draw any arbitrary line C D, and 

 apply at opposite extremities of it, 

 two equal aud opposite forces Q,, such 



that Q, -CD = P, -A, B, or Q t - P, -- 1 - Tlu-u 



Prop. XIV., the couple 

 the couple Q, 'CD. 



- 

 P a 'A, B, may be replaced by 



Fig. 55 (1). 



HN HM-oo. 

 nM' 211 N H Moos. NHM. 



and 



Similarly apply at C and D two additional 

 opposite forces Q 3 such that Q 2 = P.. , . | , 



The couple P a 4 A 2 B a may be replaced by the couplo 



"Also the couple P 8 'A, B s may be replaced by the 

 couple Q, -C D, provided Q s = P s , , , 



If one of the couples as P. 'A 4 B 4 act in the opposite 

 direction to the others, it will be replaced by the couple 



A B 

 Q 4 -C D, where Q 4 P 4 ~Q^~ 



The force Q 4 acting in the opposite direction to the 

 forces Q,, Q, and O v 



Hence, on the whole, wo have replace<l the fnr 

 couples P, -A. B,, P, -A 2 Bj, P s "A, IV. l\ \, B t , 

 acting in any din-rtinn on a rigid body in the name plaaa 

 by four couples, acting at the extremities of the same 

 arm. 



