Line of swiftest Descent. 225 



body employs less time in descending along a small arc of a 

 circle of which the inferior tangent is horizontal, than it would 

 employ in falling through the diameter ; and since the time re- 

 quired to pass through the diameter is the same with that re- 

 quired to describe any chord AB, it will be seen, that a body 332 - 

 would pass sooner from A to B, by descending along the arc AB, 

 than by moving through the straight line AB. Therefore, al- 

 though the straight line is indeed the shortest way from one point 

 to another, it is not that which requires the least time for the pas- 

 sage of a heavy body. 



Of the Line of swiftest Descent. 



351. Not only is not a straight line that along which a heavy 

 body would proceed in the shortest time from one point to anoth- 

 er, out of the same vertical, but it is not the arc of a circle which 

 answers to this description ; it is the arc of another curve which 

 may be found in the following manner. 



Suppose AMR to be the curve sought, or that through which Fig.166, 

 a heavy body would pass in the least time from a given point A, 

 to a given point B. If we take in this curve two points M, w', 

 infinitely near to each other, the arc Mm' must also be described 

 in less time than any other arc passing through these same points 

 jlf, ra', since these two points may be taken as the very points in 

 question. Having taken the point JV* infinitely nearer to Mm' 

 than M is to ra', suppose infinitely small straight lines MN, Nm* 

 to be drawn ; since the time of describing M m m' must be a 

 minimum, it follows that the difference between the time of pass- 

 ing through Mm ml and the time through MN m!, which is the 

 differential of the time, must be zero. 



Through the points JW, JV, m', draw the horizontal lines MP, 

 m P', m' P", and through A, the vertical line AC. Call AP, x, 

 PM, y ; AM, s, and suppose M m = m m x , or that, d s is constant. 

 Then mr = dx, rM = dy, m/ = dx-\-ddx, r 1 m! = dy+ddy. 

 Let u be the velocity with which the body describes Mm; it will 

 be the velocity with which MN is described; and u -\- du will 

 be that with which mm! and N m' will be described. Therefore 

 Mech. 29 



