Skating on Thin Ice. 17 



Introducing the condition 



ax ax ax 



— (773 cos a — 774 sin a) sin A + (773 sin a + 77^ cos a) cos A = 0, 



773 sin (« — A) + 774 cos (a— A) = 0, . . . (44) 



. 7 N 774 ch at sin />/ tan 6/ e . KN 



tan(A-a) = -^ = -t— 1 zj = -ivr-7» • ■ • C 45 ) 



y 773 sn at cos ot tn a/ 



sin (A — a) _ cos (A — a) __ 1 . .«. 



chaZsinfi/ sha/cosfr/~~ t/(2 . ch 2al — cos 26/), 



773= <v/(i-ch 2a/— cos 26/) cos (A — a), 



774 = x/Ci' 00 2a/— cos 2bl) sin (A— a), . . (47) 



.■at ,r=+/. 



The extra biiO) r anc} r over the length 2/, required for the 

 load W, is given by 



W = I —pr)dx •=■ -.— - I (crji sin A — cr} 2 cos li)dx 

 J -1 ^ c ^ 



*)( i 



= -jJ~ \_(vz cos ot+ij 4 sin a) sin A— ( — 77,; sin a 4-774 cos a) cos A] 



= -g^ [fjssin (A + a) — 774 cos (A-fa)] 



= rpsin 2a ^ (2. ch 2al— cos 26Z), (48) 



which determines C. 



14. If the extra buoyancy of the water is taken into 

 account in Bevan's experiment of § 8, where the tongue o£ 

 ice is depressed into the water by the weight applied at the 

 free end, we take 



v = B J T7 1 +B 2 ii2+B3% + b 4 *7 4 , .... (1) 



e con 

 -a = b, 



CbTI 



and the condition 77 = 0, ~ — 0, at # = 0, requires, since 



B 1= =0, B 3 + B4 = 0, 



77=B 2 77 2 -fB 3 (773-77 4 ) (2) 



>M. il%. S. 6. Vol. 31. No. 181. Jan. 1916. C 



