356 Quinby on the maximum effect of Machines. 
obvious that no such case can va occur in practice. ‘This 
formula, therefore, is of no value. 
The second problem Mr. Whewell gives is 
P raises g by means of a wheel and axle, as in art. 93; 
the axle being given, to find the wheel, that the time of ¢g as- 
cending through a given space, may be the least possible, 
D The accelerating force on q is 
he (Pa—qb) gb 
Pa? + qb? +MK2 
And, as this is constant, we have ag , which will mani- 
_-festly be least when f is greatest. Therefore, we must have 
oe = max. 
Pa? +96? + MK? 
If we suppose a to vary, K will also vary in a manner de- 
pending on the form of the. wheel; but if we suppose M to 
be small, we have, neglecting it, 
and seg ts Seppe a variable, 
+ qb*)—2Pa(Pa—qb)=0 ; 
“Pas —2qab—qb* =o; 
=qb ; lev(l+") ; ‘ 
If P be small compared with q, this will give nearly 
2qb , b 
é=—= ae 
Pp ts 
The weight P must act ata oi more than twice the dis- 
tance at mee it would balance q. 
This Prob. is different from both of those which we have 
before considered. The same error, however, is embraced 
in the expression here given, as has been pointed owt in the 
expression given by Dr. Gregory and Prof. Farrar. F 
Ao K in age formula is the radius of gyration. See Whewell, p. 235. 
_ Pa’g 
.. =e 
‘Pa? +MK?~ 
is the same as that given by Dr. Gregory in his 3d corollary. 
