Brachystochrunous Course of a Ship. 35 



cisely the same as that given above, and leading to the same 

 equation (A). Hence this equation is satisfied by p = 0, 

 and consequently C = 0. Hence the equation, 



1 _ 2/' (/>« )*/ 1+7 _ 



or, f(p*) - 2f f/) (1 + p*) = 0, (B) 



belongs to the straight lines which answer the condition of 



minimum we are seeking. It appears from the nature of the 



equation (B), that if any positive value of p satisfy it, an equal 



negative value corresponding to the supplementary angle will 



also satisfy it. 



1 dV 

 Let/(7; = V; then/'(^) = — . — . Substituting 



these values in (B), there results, V p dp—d V (1 + p 2 ) = 0, 



d s 2 

 which gives by integrating, V 2 = C (1 + p~) = C. -.— ; ,,. Con- 



dx 1 

 sequently j-^ = C, showing that the resolved part of the 



velocity parallel to the direction of the wind is always the same. 

 This will be the case if the direction of the ship's course al- 

 ways make a given angle, or the supplement of that angle, 

 with the direction of the wind. In fact, if we take any point 

 Fig. 1. Fig. 2. 



H in AE, the shortest course from H to F must be in HE, 

 EF, otherwise AE, EF is not the shortest course from A to F. 

 Therefore when the bearing of the two places from each other 

 is altered, the angle which the ship's course makes with the 

 direction of the wind remains the same. If the point e be 

 very near to E, the shortest course from e to F is principally 

 in EF, which differs but little from the straight line joining e 

 and F. Hence the angle which EF makes with OW is that 

 which is made by the ship's course with the direction of the 

 wind, when it first begins to be of advantage to tack instead 

 of proceeding in the direct course between the two places. 



F 2 



