422 Mr. O'BRIEN, ON A NEW NOTATION FOR VARIOUS EQUATIONS 



And, by the parallelogram (or rather, the polygon) of forces, these are equivalent to the three 

 forces, 



2P at p, SQ at q, ^K at r. 



Hence the conditions of equilibrium of these three forces are the conditions of equilibrium of the 

 forces U, U\ U", &c. Therefore, by Lemma 2, the conditions of equilibrium of the forces U, U', 

 U", &c. are 



2P+2Q + 2i? = (32). 



£»;> . 2P + Z>g . SQ + Dr . 2ff = (33). 



Now, since U is tlie resultant of P, Q, and R, we have 



P+Q + R= U, 

 and therefore (32) becomes, 2 C7 = 0. 



Also we have 



Du.P + Du.Q + Du. R ^ Du . U, 



and therefore, by Lemma 1 , 



Dp.P+ Dq.Q + Dr.R = Du .U. 



Hence (33) becomes '2Du . U = 0. 



It appears therefore that the necessary and sufficient conditions of equilibrium of the forces 

 U, U\ U", &c. acting at the points ?«, ii, u," &c. of a rigid body, are 



2C/ = (28). 



*'LDu.U = (29). 



(HI.) 



23 The equation (29) includes the whole theory of couples. 



For, suppose the forces U, U\ U", &c. to constitute a set of couples, in other words, suppose, 

 that 



U'= - U, U"'= - U", &c. &c. 

 Then the equation (29) evidently becomes 



D (11- u) .U + D (?/"- u") . U"+ &c. = (34), 



Now, by Art. (5), if r and R be tlie numerical magnitudes of m' — u and U, and 6* the angle 

 contained by n' — u and U, then the numerical magnitude of D {u - u) . U is Rr sinfl; which is the 

 moment of the couple consisting of U and If; for r sin t* is evidently the perpendicular distance 

 between IT and U' . Also D (u'— u) . f/ is a line perpendicular to u' — u and U, and therefore to 

 the plane of the couple {U, IT"). Hence D {u - u) . U is the a.vis of the couple (U, U'). 



The equation (34) therefore indicates, that the symbolical sum of the axes of a set of couples wliicii 

 balance each other must be zero. Which includes all the propositions of the theory of couples*. 



(IV.) 



24. When the forces U, U', U", &c. do not balance each other, to find the condition of 

 their having a single resultant. 



Suppose that R is the resultant, and r its point of application ; then since — R, U, U', &c. 

 balance each other, we have, by (28) and (29), 



2U-R = 0, 'S.Du.U - Dr.R = 0, 



" Respecting this equation, we sliould have remarked, that Du.U is the symbol of the axis of the couple which transfer.-- the 

 force U from the point « to the origin. See Article 24, page 423. 



