PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 297 



plane applied to a body moveable about a fixed axis, the direction of one of them P^ 

 passing through the centre of the axis, and the system being in the state bordering 

 upon motion by the preponderance of Pj*. 



Let /i 2 *i3 ^2-} t)e taken, as in the preceding section, to represent the inclinations of the 

 directions of the pressures Pj, Pg, P3 to one another, and a^, Og the perpendiculars let 

 fall from the centre of the axis upon P^, Pg; and X the perpendicular let fall from the 

 same point upon the resultant R of Pj, Pg, P3. Then since R is equal and opposite to 

 the resistance of the axis (section 8.), and that P3 acts through the centre of the axis, 

 and Pi and Pg act to turn the system in opposite directions about that centre, 



Pi ^1 — P2 02 == ^ ^' 

 Substituting for R its value -J-, 



Piai-P2a2 = ?^{Pi'4-P2' + P3' + 2PiP2COSii.2 + 2PiP3COS/i3 + 2P2P3C08i23}*; 

 squaring both sides of this equation and transposing. 



Pi' W - ?^2) « 2 Pi { P2 fli a, + X2 (P2 cos /i^ + P3 cos /I j) } 

 = - F.^aj^ + X2 |P^2 + P32 4. 2 P2P3 cos/23}; 

 solving this quadratic in respect to P^ and omitting terms which involve powers of 

 X above the first, 



Pi ai2 = P2 Oi ^2 + ?^ {P2' W + 2a,a2 cos t,^ + a^^) + P32 0^2 



+ 2 P2 P3 «i (^2 cos /i3 + fli cos ^23) }* ; 



or representing the line which joins the feet of the perpendiculars a^ and Oj by L, and 

 the function ^i (ag cos i^^ + ^1 cos I22) by M, 



Pl=P2(|f)+^«{P2'L2 + P32ai2 + 2P2P3M}* (12.) 



If P3 be so small as compared with Pg, that in the expansion of the irrational quan- 



p, 

 tity, terms involving powers of p^ above the first may be neglected, the above equa- 

 ls 



tion will become by reduction, 



p. = (I:){'+^}p. + ?lP3 (.3.) 



If in the expressions represented by Lj 2 and M23 (section 9.) we make a.^ = 0, 

 give to flg ttic negative sign (since the forces Pi and P2 tend to turn the system in 

 opposite directions about the axis), and observe that, since Pg receives an opposite 

 direction, cos /2„3 becomes negative J, these expressions will become identical with 

 those represented by L and M in the preceding equation (12.), and that equation will 



* This problem is here investigated by an independent method as a verification of the theorem established in 

 the preceding article, and as an application of it to a case of frequent occurrence in machinery, 

 f PoissoN, M^canique, Art, 33. J See note, p. 295. 



