300 



PROFESSOR MOSELEY ON THE THEORY OP MACHINES. 



Now C^ being essentially positive, this quantity is a minimum when 



2 «! ^2 ^2 (U2 COS /i,2 + P3 S2 COS l^^^) 



is a minimum ; or, observing that U2 = P2 Sg, and dividing by the constant factor 



2aia2U2^2'^ when 



Pg cos /j 2 + P3 cos ti2 is a minimum. 



From the centre of the axis C let lines C piC p2 be drawn parallel 

 to the directions of the pressures Pj, P2 respectively; and whilst Cp2 

 and C P3 retain their positions, let the angle p^ C P3 or i^^ be con- 

 ceived to increase until P^ attains a position in which the condition 

 P2 cos I12 + P3 cos ii2 = a minimum is satisfied. Now 



PiCV.^=p^Cp2-p2C P3, or /1.3 = /i 2 - '2.3 ; 

 substituting which value of /j^, this condition becomes 



P2 cos /j 2 + P3 cos (/i 2 ~ '2.3) = a minimum. 



or 



or 



Let now 

 so that 



Pg cos t^2 + P3 cos ii2 cos ^23 + P3 siu /^ 3 sin /2.3 = a minimum, 

 (P2 + P3 cos /2.3) cos /i^ + P3 sio ^2^ sin /j g = a minimum. 

 Pssini a^ = tan ^ 



Pg + Pg COS «23 '' 



P3 sin ;2;j = (P2 + P3 cos I22) tan y 

 .*, (Pg + P3 cos /2.3) COS /j 2 + (P2 + P3 COS t2^) tan y sin /j 2 = a minimum, 

 or dividing by the constant quantity (P2 + P3 cos /g^), and multiplying by cos y, 

 cos ii2 cos y + sin /^ 2 sin y = cos (/i 2 — y) = a minimum. 



•1.2 



— ty =: T. 

 Pg sin <2; 



, f PgSiriigo "1 



.-. .,2 = . + tan-i|p^^fp^^J 



p^ + Pgcos*--^ r (^9.) 



To satisfy the conditions of a minimum, the angle p^ C p2 must therefore be in- 

 creased until it exceeds 1 80° by that angle y whose tangent is represented by 



Pg sin ig^g 



P2 + Pg COS I2.3 



To determine the actual direction of P^, produce then p2 C to q, make the angle q Cr 

 equal to y ; and draw C m perpendicular to C r, and equal to the given perpendicular 

 distance a^ of the direction of Pj from the centre of the axis. If m P^ be then drawn 

 through the point m parallel to C r, it will be in the required direction of P^ ; so that 

 being applied in this direction, the moving pressure Pi will work the machine with a 

 greater economy of power than when applied in any other direction round the axis. 



It is evident that since the value of the angle 1^2 or /?oCjt?i, which satisfies the con- 

 dition of the greatest economy of power, or of the least resistance, is essentially greater 

 than two right angles, Pj and P^ must, to satisfy that condition, both be applied on 



