300 
PROFESSOR MOSELEY ON THE THEORY OF MACHINES. 
Now C 2 being’ essentially positive, this quantity is a minimum when 
2 Cly U 2 (^2 ^OS *1.2 ”1“ ^3 ^2 ^OS h. 3 ) 
is a minimum ; or, observing that XJ 2 = P 2 S 2 , and dividing by the constant factor 
2 a x a 2 U 2 S 2 ; when 
P 2 cos q 2 J r P 3 cos q 3 is a minimum. 
From the centre of the axis C let lines C p l C p 2 be drawn parallel 
to the directions of the pressures P l5 P 2 respectively; and whilst C p 2 
and C P 3 retain their positions, let the angle p x C P 3 or q 3 be con- 
ceived to increase until P x attains a position in which the condition 
P 2 cos q 2 + P 3 cos q 3 = a minimum is satisfied. Now 
Pi C P 3 = p l c p 2 - p 2 C P 3 , or q 3 = q 2 - < 2.3 ; 
substituting which value of i 13 , this condition becomes 
or 
P 2 cos q 2 + P 3 cos (q 2 — < 23 ) = a minimum, 
or 
P 2 cos q 2 + P 3 cos q 2 cos i 23 + P 3 sin q 2 sin < 2.3 = a minimum, 
(P 2 + P 3 cos / 2 . 3 ) cos q 2 + P 3 sin i 23 sin q 2 = a minimum. 
T Po sin o 
Let now p 8 +p,co Sto =tan y’ 
so that 
P 3 sin q ^ = (P 2 + P 3 cos i 23 ) tan y 
.*. (P 2 + P 3 cos < 23 ) cos q 2 + (P 2 + P 3 cos i 23 ) tan y sin q 2 = a minimum, 
or dividing by the constant quantity (P 2 + P 3 cos i 23 ), and multiplying by cos y, 
cos q 2 cos y + sin q 2 sin y = cos (q 2 — y) = a minimum. 
' 1.2 — V = k. 
, f P, sin 1 . 
q .2 = * + tan- | P 2 + p 3 cos (2 3 j ( 19 ’) 
To satisfy the conditions of a minimum, the angle p 1 C p 2 must therefore be in- 
creased until it exceeds 180° by that angle y whose tangent is represented by 
P 3 sin < 2 3 
P 3 + Po cos I 
2.3 
To determine the actual direction of P l5 produce then p 2 C to q, make the angle q C r 
equal to y ; and draw C m perpendicular to C r, and equal to the given perpendicular 
distance a x of the direction of P x 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 x ; so that 
being applied in this direction, the moving pressure ?! 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 q 2 or p 2 Cp x , 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 2 must, to satisfy that condition, both be applied on 
