Solutions of Problems in the Calculus of Variations. 197 



problem and that which my solution exhibits. Mr. Todhunter 

 assumes a certain discontinuity of the joining line, after which it 

 becomes a very simple matter to obtain a solution ; whereas in 

 the process which I have employed that particular discontinuity 

 is inferred, I make this remark expressly for the purpose of 

 raising the question as to whether the solutions of problems in 

 the Calculus of Variations may involve the principle of discon- 

 tinuity, This being a question of a novel and interesting cha- 

 racter, I propose now to discuss it at some length, and to give 

 my reasons for answering it affirmatively. 



Before entering upon considerations more directly bearing 

 upon the subject for discussion, I beg to advert to an assertion, 

 noticed by M. Lindelof, and which I cannot but regard as un- 

 scientific, to the effect that the Calculus of Variations fails to 

 give a solution of the problem above enunciated. Upon the 

 general principles of analytical calculation it may, on the con- 

 trary, be asserted that, if there be in rerum naturd a quantitative 

 maximum or minimum, there must be appropriate rules by which 

 its amount may be calculated either exactly or approximately. 

 We may fail in our knowledge of the rules ; but calculation can- 

 not be said to fail. This general remark may be illustrated by 

 the following example. 



It is known, from the practice of navigation, that the bra- 

 chystochronous course of a sailing-vessel from one point to an- 

 other is generally a broken line. How, it may be inquired, does 

 analytical calculation determine such a course? Let A' and|B' 

 be the two points, A' being the point of starting ; and let this 

 point be also the origin of coordinates. Then if the axis of x be 

 in the direction in which the wind blows, and p be the tangent 

 of the angle which the ship's course at any instant makes with 

 that axis, the rate of sailing is some function of this ; angle, and 

 therefore of p. As, however, the rate is the same, whether the 

 direction of sailing makes a certain angle, or its supplement, 

 with the axis of x, the wind in the two cases striking on oppo- 

 site sides of the ship, we shall have 



rate of sailing =/(^ 2 ), 



which equation expresses all the conditions of the problem. 

 Hence the quantity to be a minimum is 



J 



' 1+ ^ 



The differential equation given by the rules of the Calculus of 

 Variations for this instance is 



v\+ff{p*) {f{p*)Y 



