232 Prof. Morton and Mr. Tobin on Times of Descent 



,\ p=z±<j(t' — t) 2 , vanishing, us it obviously should, for 

 the locus of which the straight line forms part. 



Fig. 3. 



For times of descent shorter than that along the line the loci 

 are detached from B and form loops which contract to a point 

 corresponding to the minimum time from B to along a 

 two-line path. In the case where the two end-points are on 

 the some level (fig. 5), it is obvious from elementary 

 considerations that the two lines are inclined at 45° to the 

 horizontal. 



We now examine the general case of the " two-line brachi- 

 stochrone." 



Let the inclinations of OA, OB be u ft and their lengths 

 a b, and let c, y be length and inclination of AB. Keeping 

 b (3 fixed we require two equations satisfied by <xy when the 

 time along BAO is a minimum. These can be got by 

 equating to zero the partial differentials of the expression for 

 the time in terms of « /3 y and b. But they are obtained 

 mucli more neatly by considering geometrically the effect of 

 displacing A in two special directions. 



