100 MECHANICAL RECREATIONS [CH. V 



to take one instance, the sail to be fixed, that is, suppose a to 

 be a constant. Then v is a maximum if 8 + a = \tt, that is, if 

 is equal to the complement of a. In this case we have 

 v = u cosec a, and therefore v is greater than u. Hence, if the 

 wind makes the same angle a abaft the beam that the sail 

 makes with the keel, the velocity of the boat will be greater 

 than the velocity of the wind. 



Next, suppose that the boat is running close to the wind, 

 so that the wind is before the beam (see figure below), then 

 in the same way as before we have v sin a = u sin (6 + a), 

 or v sin a = u sin <p, where $ = angle WAS = ir— — eu Hence 

 v = u sin <f> cosec a. 



Let to be the component velocity of the boat in the teeth 

 of the wind, that is, in the direction AW. Then we have 

 w = v cos BA W = v cos (a + <p) = u sin $ cosec 2 cos (a + <£). If a 

 is constant, this is a maximum when <\> = \tt — Ja; and, if <f> 

 has this value, then w = \u (cosec a — 1). This formula shows 

 that w is greater than u, if sin a < £. Thus, if the sails 

 can be set so that a is less than sin -1 £, that is, rather less 

 than 19° 29', and if the wind has the direction above assigned, 

 then the component velocity of the boat in the face of the 

 wind is greater than the velocity of the wind. 



The above theory is curious, but it must be remembered 

 that in practice considerable allowance has to be made for 

 the fact that no boat for use on water can be constructed in 

 which the resistance to motion in the direction of the keel 

 can be wholly neglected, or which would not drift slightly to 

 leeward if the wind was not dead astern Still this makes less 



