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PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
Pi a \ + ?2 a 2 + P3 a z H“ • • ^ 
Pi 2 + p 2 2 + P3 2 + 
+ 2 Pj P 2 cos q 2 + 2 Pj P 3 cos q. 3 + . 
+ 2 P 2 P 3 cos q 3 + 2 P 2 P 4 cos q 4 + 
+ &c. &c. 
Pi = - 
P 2 « 2 + P a a s + 
3 “ 3 ' 
~ + 
Pj 2 + 2 P 4 (P 2 cos q 2 + P 3 cos < L3 + ) ' 
+ P 2 2 + P 3 2 + P 4 2 + . . . . . 
+ 2 P 2 P 3 + 2 P 2 P 4 + 
+ &c. &c. 
If the value of P 4 involved in this equation be expanded by Lagrange’s theorem*, in 
a series ascending by powers of X, and terms involving powers above the first be 
omitted, we shall obtain the following value of that quantity : — 
-f P 3 « 3 + P 4 a 4 + ) 2 
T) P 2 « 2 + P3 «g + . . . 
r l — ~ n . 
< ( p : 
2 2 
+ (t)' 
a x (P 2 a 2 + P 3 a 3 + P 4 a 4 + )• 
(P 2 COS q 2 + P 3 COS q. 3 + P 4 COS q >4 + ) 
+ p 2 2 + p 3 2 + p 4 2 + . ...... 
+ 2 P 2 P 3 cos i 23 + 2 P 2 P 4 cos i 24 
+ 2 P 3 P 4 cos / 3 . 4 + ..... 
or reducing, 
Pi = ~ 
Po da + P 3 do + 
4 
P 2 2 (tq 2 -2fl 1 « 2 cos q 2 -f- ^ 2 2 ) 
+ P3 2 («1 2 - 2 a \ «3 COS q 3 + « 3 2 ) 
+ &c. &c. 
-f 2P 2 P 3 {a 2 a 3 — a x (<q cos/ 23 4-a 2 cosq 3 + a 3 cosq 2 )} 
-f-2P 2 P 4 {<z 2 £q — <q(<q cos / 24 ~\~cl 2 cos q , 4 + a 4 COSq 2 )} 
-j- &c. &c. 
Now <q 2 — 2 <q a 2 cos q 2 + a 2 2 represents the square of the line joining the feet of the 
perpendiculars <q and a 2 let fall from the centre of the axis upon P 4 and P 2 ; similarly 
tq 2 - 2 «j a 3 cos q 3 + a 3 2 represents the square of the line joining the feet of the per- 
pendicular let fall upon P 4 and P 3 , and so of the rest. Let these lines be represented 
by Lj 2 , L 13 , Lj 4 , &c., and let the different values of the function 
{a 2 a 3 — a x {a x cos / 23 + a 2 cos q 3 + a 3 cos q 2 ) } 
be represented by M 23 , M 24 , M 34 , &c., 
.-. P x = 
P 2 « 2 + P 3 « 3 + . . . x_ f P2 2 L 12 2 + P3 2 L l3 2 -j- P 4 2 L l4 “+ . . 
a, 3 1 
+ 
(no 
+ 2P 2 P 3 M 2 . 3 + 2P 2 P 4 M 24 + . 
10. The conditions of the equilibrium of three pressures P,, P 2 , P 3 in the same 
* This expansion may be effected by squaring both sides of the equation, solving the quadratic in respect to 
P, , neglecting powers of A above the first, and reducing ; this method is however exceedingly laborious. 
