248 On the Measure of 
it is raised in the least time, P > must’ 
descend ‘through a space =AB, but when 
it is raised with the least moving force, 
P descends through a space =1AB only. 
For, if we make BD=ZAB, and let W ascend: 
along any concave surface DEB, of which 
BD is the ehord, it will be raised to B by 
the descent of P through a space =BD, and 
it will be at rest when it arrives. at B. This 
is so obvious, that it would be superfluous to 
give a demonstration of it. 1t appears then, 
that ‘twice the quantity of moving force which 
is absolutely necessary to raise W to B, must 
be expended if it is to be raised by P im the 
least time. To determine the curve by which 
W will ascend from D to B in the least time, 
is an intricate’ problem, and I do not know 
that it has ever Leen solved ; but a practical 
approximation to it in any particular case may 
be easily found. A well constructed steam- 
engine for raising water exhibits in’ every 
stroke a practical example of the same pro- 
blem. At the commencement of the stroke, 
a very great pressure of steam is thrown upon | 
the piston, and this pressure is gradually 
diminished, so that at the end of the stroke 
there is a considerable preponderance in the 
opposite direction. In consequence of this 
