296 



PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 



Pl«l + P2«2 + P3«3+---- = ^ 



+ 2 Pi P2COS/1.2 + 2 P1P3COS /1.3 + 



+ 2 P2 P3 COS Ii3 + 2 P2 P4 COS /2.4 + 



+ &c. &c. 



Pl2 + 2 Pi (P2 COS /1.2 + P3 COS /1.3 + ) ^ 



+ P22 + P32 + P/+ 



+ 2P2P3+2P2P4 + 



. + &C. &C. 



If the value of Pi 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 : — 



.•.Pi = - 



Po «o + Po Oo + . . . 



+ 



Pl=- 



Po flo + Po 0.0+... 



+(^) 



^(P2«2+P3«3 + P4«4 + y 



2 

 "- ^ (P2 «2 + P3 «3 H- P4 «4 + )• 



(P2COS/i2+P3COS/i^+P4COS/i4-f ) 



+ p^2 4. p^2 4. p^2 + 



+ 2 P2 P3 COS /2.3 + 2 P2 P4 COS /2.4 

 + 2P3P4COS/3.4+ 



or reducing, 



Pi = - 



Pjfl, + Ps<»3 + .. 



^h 



V^ («i^ — 2 «! «2 COS /i 2 + a^) 

 + P32 (ai2 - 2 fli fl3 cos /i,3 + a^) 

 + &c. &c. 

 + 2P2P3{a2«3—«i(«i COS /23+a2Cos/i3+a3 cos/12)} 



4-2P2P4{a2«4-«lKcOS/2.4 + «2COS/i4+a4COSii^)} 



+ &c. &c. 



Now a-^ — 2 tti ^2 COS i^^ + ^2 represents the square of the line joining the feet of the 

 perpendiculars a^ and a^ let fall from the centre of the axis upon Pi and P2 ; similarly 

 a^ — ^a^a^ cos ^i 3 + a^ represents the square of the line joining the feet of the per- 

 pendicular let fall upon Pi and P3, and so of the rest. Let these lines be represented 

 by Li 2, Li 3, Lj 4, &c., and let the different values of the function 



{a^ ^3 -- «! (fti cos 1^2 + «2 COS ^13 + «3 COS /j 2)} 



be represented by Mg^, M24, M34, &c., 



P9«9 + P3«3 + .. 



. p ^ 2 "■a T- ^ 3 "^3 T • ' • i_ A f 



• • *^1 — - n- -r «j2 \ 



X ; P22 Li/ + P32 Li/ 4- P4^ Li,2+ . . . 



+2P2P3M23+2P2P4M2.4+ 



:} 



(11.) 



10. The conditions of the equilibrium of three pressures P^ P2, P3 in the same 



* This expansion may be effected by squaring both sides of the equation, solving the quadratic in respect to 

 Pj, neglecting powers of X above the first, and reducing ; this method is however exceedingly laborious. 



