



Skating 



on 



Thin Ice, 



as before in 



(i) § 



2, with 







and with 





<7 



I 



= 1,3. 



12 ' 



U 2 = ,l ,, , = ^^ , (10) 



1 + methmh ge , 2 , v 



l + ^th 2 ??i/i 



in which th?>i/i may be replaced by unity, when the depth h 

 of the water is more than the wave-length X, making, as 

 in §2, 



U 2 = V 7 (11) 



1 + ^ 



5. The determination of the exact minimum value of TJ 2 

 in (10) § 4 will lead to an intractable quintic equation, even 

 with thm/i replaced by unity in (11) ; but as the term ge/Y 2 

 in the denominator is small, we may consider the variation 

 of IP as due to the numerator alone, with thm7* omitted as 

 unity, 



U 2 = V 2 + l(fi)V, . . (i) 



and this is a minimum by variation of V when 



V 2 = J-(f 2 )V, (2) 



making 4 TJ 



S IP = |V 2 , ^=M5, 



that is, XJ about 15 per cent greater than V. 



This is in accordance with the simple theorem that the varia- 

 tion of ax mj \-hx~ n is zero when its d.c. max m ~ l — nbx~ n - J = 0, 

 or max m =nbx- n , and here m=l, n=3. 



As another example, the cost per mile of a steamer at 



A 



K knots being given by the two terms -^ + BK 2 , it is a 

 minimum when ^ = 2BK 2 , or with the running expenses 

 and wages twice the coal bill. 



