562 Mr. John Wylie on Graphical 



and C^Bg these being the tangents to the circles, and touching 

 at D 1 and D 2 . 



Time from A, to C 2 = ^ + ?A 



Let C/ be a point in the common edge very near to C 1? so 

 that ~DiCi and Ci'D 2 can be taken as the steering directions 

 for path A 1 C l , C 2 . 



When A^C^ is a minimum time path the times along 

 these two paths will be the same, 



A^/ , C 2 B 2 ' _ A,B, C 2 B 2 



Vi v 2 v l v 2 



BiB/ B 2 B 2 ' 



(1) 



Drop perpendiculars Di^Mx and D 2 N 2 M 2 on the edges of 

 the belts. 



Then by similar triangles 



BJB/ : CA' = DxNx : D^, 



„ ■ — ai sin u x 



•*• BxB/ = CiCi x jy-jyj- = CiCi x 



— ai sin a x 4- «i 



= CA' 



U 

 »l + - 



sin «x 

 Similarlv, 



B 2 B 2 ^GiCi — 

 Substituting into (1) gives 



^2 



u 



sin a 2 



tfi+- = v 2 -f- , (2) 



sin «! sm a 2 K y 



which is the condition to be satisfied for a minimum time 

 path. 



This condition is independent of the breadths of the belts. 

 It may be called the " Law of Refraction v as regards the 

 steering directions in passing from one belt to another 

 flowing at a different velocity. a x and a 2 being the angles 

 of incidence and refraction. 



It follows at once that for any number of parallel belts 



U U u a rr 



t'i+ -s =v 2 + =v 8 + -i = &c. = K say, 



sin a\ sin «<> sin uz J 



