Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A cycloid; the curve of fastest descent between two points.
from The Century Dictionary and Cyclopedia
 n. The curve upon which a body moves in the least possible time from one given point to another.
Etymologies
Examples

It was almost a dramatic scene that day when he leaped from his chair and scrawled the equation of the "brachistochrone" on the blackboard in a handwriting that no mortal man could decipher.
In the days of my youth when I was a student in the University of Virginia, 18881893.

The brachistochrone curve that gives the minimum travel time is determined by using the calculus of variations to find the best compromise between the shortest path – i.e., a straight slope – and the path that gives the maximum initial acceleration, i.e., a vertical drop followed by a short bend followed by a horizontal path.

So again, as in the case of the brachistochrone, there is a reason for minimization, but I do not see any reason for minimization in the case of the proton.

Another is the famous brachistochrone problem in which a ball rolls down a curve a cycloid that gives the minimum time of travel.

The brachistochrone problem is different because there the rolling ball follows a constrained path.

To understand the true relation of these theories in that part of the field where they seem equally applicable we must look at them in the light which Hamilton has thrown upon them by his discovery that to every brachistochrone problem there corresponds a problem of free motion, involving different velocities and times, but resulting in the same geometrical path.

Finally, the cycloid is also said to be the brachistochrone (from the Greek brakhistos, or shortest, and khronos, or time) because it is the path of an object falling freely from the ﬁ xed point A to the ﬁ xed point B in the shortest possible time.

My Sentence Shelby Donald stood among the remains of last night's argument, the third in as many days, when she realized she was caught in a brachistochrone problem of no small significant for no matter how loudly she availed upon the God's of better judgement her inbreed intemperance demanded satisfaction and would always gravitate her towards the most mercurial of men.

DFW's Sentence The closest conventional analogue I could derive for this figure was a cycloid, L'Hopital's solution to Bernoulli's famous brachistochrone problem, the curve traced by a fixed point on the circumference of a circle rolling along a straight line.
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