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 3 
passing through the centre of the axis, and the system being in the state bordering 
upon motion by the preponderance of Pj*. 
Let q 2 <i 3 <23 be taken, as in the preceding section, to represent the inclinations of the 
directions of the pressures p n P* P 3 to one another, and a x , a 2 the perpendiculars let 
fall from the centre of the axis upon P 1? P 2 ; and X the perpendicular let fall from the 
same point upon the resultant Rof P l3 P 2 , P 3 . Then since R is equal and opposite to 
the resistance of the axis (section 8.), and that P 3 acts through the centre of the axis, 
and P 2 and P 2 act to turn the system in opposite directions about that centre, 
Pj a x — P 2 a 2 = X R. 
Substituting for R its value -f-, 
P 1 a \ — P 2 <h = x { p i 2 + P 2 2 + p 3 2 + 2 P ! P 2 cos q 2 + 2 Pj P 3 cos #j 3 + 2 P 2 P 3 cos / 2<3 }* ; 
squaring both sides of this equation and transposing, 
P, 2 («1 2 — X 2 ) - 2 P, {P 2 a, a, + X 2 (P 2 cos I 12 + P 3 cos i u ) } 
= - P 2 2 « 2 2 + X 2 {P 2 2 + P 3 2 + 2 P, P 3 cos < 23 }; 
solving this quadratic in respect to P 1? and omitting terms which involve powers of 
X above the first, 
p i a \ — p 2 a \ a 2 + x ( p 2 2 K 2 + 2 a x a 2 cos < 12 + a 2 2 ) + P 3 2 a 2 
+ 2 p 2 P 3 «! (a 2 cos q. 3 + «i cos < 23 )}^ ; 
or representing the line which joins the feet of the perpendiculars a x and a 2 by L, and 
the function a x (a 2 cos < 13 + a \ cos < 23 ) by M, 
Pi 
+ {P 2 2 V + P3 2 a? + 2 P 2 P 3 M}*. . 
• • ( 12 .) 
If P 3 be so small as compared with P 2 , that in the expansion of the irrational quan- 
p 
tity, terms involving powers of above the first may be neglected, the above equa- 
*2 
tion will become by reduction, 
p . = (^){'+^} P2 + ^ i P2 < 13 -) 
If in the expressions represented by L L2 and M 23 (section 9 .) we make « 3 = 0, 
give to a 2 the negative sign (since the forces Pj and P 2 tend to turn the system in 
opposite directions about the axis), and observe that, since P 2 receives an opposite 
direction, cos ; 23 becomes negative 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, Mecanique, Art. 33. \ See note, p. 295. 
