VOL. XLIX.] PHILOSOPHICAL TRANSACTIONS. 55Q 



-r = (^ "'" i") '^' ^* "^^^ ^^^ constant ratio of the pressure to the friction be 



that of 1 to /* ; then the friction will be ( 1 + -) ttjw, ; and the momentum of this 



friction = (1 + r) CTTj/. : but the momentum of the friction from the weight of 



the machine is = mc;u ; which added to the former momentum, gives ((1 + ^) 



» + m) Cf* : hence the momentum of the moving power is=Aa — sb — ((j 



+ i) TT + m) c/x. But since the momentum of inertia is aoj'* + si* + mcP; 



Aa— Kb— ((1 -(- -) T + M) Cfo 

 therefore the accelerating force will be = — — j-—, — tt ; and for 



the acceleration of the point a, or of the moving power a, it will be, by the 



Aa* — sab — ((1 + ^) »' + m) «Cj* 



principles of mechanics, T^-TlW+Md^ X / = c; where/ denotes 



the element of the time, and c the element of the velocity. But if a should fall 

 freely, it would be - / = /. And since the increments or decrements of velo- 

 cities, generated in the same particles of time in the same body, are as the ge- 

 nerating forces, therefore >' : / — <■:: a : the force generating the decrement of 

 celerity '/ — c, which is the same force that retards the fall of the body, stretches 

 the string, and presses on the side « ; hence, substituting for the values, we 



Afl' — Ba6 — ((1 + ^) » + m) ac/b 

 have the following analogy 1:1 ^^ + bA' + m I^ :: a : tt ; there- 



ab6* + AMd* + ABOA + ((1 +^)'f + m) Aac/* 



fore IT = gi 4, b6» + Mrf* ' ^^°^ which equation is found 



abA^ + AM(f* + A BaA + A Mac/* 1 



Afl* + bA» 4- Mrf» — (1 + T)*OCIb 



wa 



ABaA +AMO' J- + ABO* + AM -j- 



— , and the whole pressure 



* Aa» + bA" + mcP - (1 + f-) AOC(i* 



AB (a + A)' + am («P + ac/i.) • ( 1 + f ) 



-l'" — 



* Aa« + bA« + Md* - (1 + A ) Aflc,* 



If now we should exclude the friction and the weight of the machine, we 

 should have the whole pressure = ^^^^- And if, as is the case in the 



.,, , AB (a + A)* 4ab 



pulley, we suppose a= b, the whole pressure will be = -^^-qrjj-^ = j:^^' 



