198 Prof. Challis on the Principle of Discontinuity in 



This equation, containing only jo and constants, shows that the 

 brachystochronous course is rectilinear ; and with this result 

 nautical practice agrees. Also it is to be remarked that the case 

 in which the wind blows directly from A' to B', in which case 

 the straight line joining these points is plainly a minimum or 

 maximum course whatever be the function f{p 2 ), must be in- 

 cluded in the general investigation. Hence it follows that the 

 equation is satisfied if J9 = 0, and consequently that C = 0. Thus 

 after getting rid of the factor p, we shall have 



/(/) -2(1 +/)/'(/) =0. 



As this result is of the form ^(j? 2 ) = 0, and is therefore equally 

 satisfied by positive and negative values of p y it is the differen- 

 tial equation of an even number of straight lines which, taken in 

 pairs, make supplementary angles with the axis of oc. The 

 values of the angles are deducible from the equation above, if 

 the form of/(^ 2 ) be explicitly given. It thus appears that ana- 

 lysis indicates, respecting the brachystochronous course, that it 

 consists of not fewer than two straight lines. But we have also 

 to satisfy the condition that the course passes through the given 

 points A' and B'. Now this may be done by means of two such 

 lines inclined at supplementary angles to the axis of x } the ship's 

 course being along one up to the point of their intersection, 

 and subsequently along the other. This result is also in accord- 

 ance with practical sailing. If it be objected that the course, 

 being on different lines, is discontinuous, we may reply as fol- 

 lows. The Calculus of Variations answers the question as to 

 the nature of the course, only so far as to give a differential 

 equation between the variables, which equation requires to be 

 afterwards solved by appropriate rules. Now it is known, from 

 the theory of such equations, that the solutions in certain cases 

 represent more than one line. Hence, if by means of more than 

 one line the conditions of a proposed problem may be satisfied, 

 and an answer k be obtained which possesses the character of 

 a maximum or minimum, there would seem to be no analy- 

 tical reason why such solution should not be accepted. In the 

 instance before us, it may be thus shown that the solution given 

 by two straight lines, determined as to direction by the foregoing 

 equation, possesses the character of a minimum. Let x and a 

 be respectively the coordinates parallel to the axis of x of the 

 point of intersection of the lines and the point B'. Then the 

 time of sailing from A' to B' is 



x^l+p 2 {a—x) Vl+p 2 as/l+p 2 



jm + M) ' or ~KfT 



