3* The Rev. J. Challis on the 



in which the ship is moving at any point of its course makes 

 with the axis of x, the velocity will be some function of 0, 

 and therefore of p, of such a nature that it will retain the 

 same value when is changed to it— 0, or, what is equivalent, 

 when p is changed to — p. We may therefore say, that the 



velocity, or ~ = f(f). Hence dt = ~~ y and 

 ±' f{P) 



4 — ^ 1 + P~ d x t> r ■ 



J ~f{p > -) ' Keternn g now to tne general equation 



for determining maximum and minimum functions according 

 to the method of the calculus of variations, viz. 



XT d(P) d 2 (Q) „ 

 N- -^— ' + -£% — &c. = 0,* 

 dx dx 1 



in the instance before us this reduces itself to -4^ = 0, be- 



dx 



cause p only is involved. Hence integrating F = C. The 

 quantity P is the differential coefficient of * 1 + P~ with re- 



„ Ap 2 ) 



spect to p. Hence 



f{ P *)Vi + f (f(p*)? " - u (A) 



This equation, as it contains only p, must belong to a sy- 

 stem of straight lines ; which proves that the brachystochro- 

 nous course is not curved, but rectilinear. Let, therefore, the 

 brachystochronous course from A to B (fig. 1.) be in a system 

 of straight lines AE, EF, FG, GB. Then the brachysto- 

 chronous course from A to F is in the two lines AE, EF. We 

 might show that these two lines make supplementary angles 

 with OW, by solving the following problem as a question in 

 maxima and minima of two variables : — Supposing the course 

 from one given point to another to be in two straight lines, 

 and the velocity to be given when the angle which the direc- 

 tion of the motion makes with a fixed straight line is given, 

 required the positions of the two lines. It would appear that 

 the resulting equations are satisfied by supposing the two lines 

 to make supplementary angles with the fixed line. But per- 

 haps the following reasoning will be deemed sufficient. The 

 case in which the wind is favourable, and blows directly from 

 A towards B, is involved analytically in the preceding investi- 

 gation, because to show that in this case the swiftest course is 

 in the straight line joining A and B, requires calculation pre- 



* See Ain's Tracts, 2nd Edit. p. 232. 



