Skating on Thin Ice, 15 



But t)x and tj 2 are absent in j? -f\i and so must be 



absent in Ai -^ -f A 2 -f^ , so that 

 dy dy' 



AiCOsf/ij — a)-f-A 2 cos(A 2 — a) and A x sin^ — a) 4- A 2 sin(A 2 — a) = 0, (22) 



At cos (h 2 — a) sin (lu — a) /7 ,. , /x N 



P = ^ ( = r-^ (, tan (fa — a) = tan (fa -a , 



A 2 cos(/i! — a) ran (fa—a)' w ■ v 



fa = A 2 = fa Ax = — A 2 = A, (23) 



A 1 ^+ A 2 ^- 2 = -2Ac Vs sin (h-*) + 2Ac Vi cos (A-a) 



= UB 1 ^+UB 2 ^ 2 

 a# a.?j 



= UB 1 c(?; 3 cos a — 7i 4 sin a) + UB 2 c(t7 3 sin a + 7/ 4 cos a), . . (24) 



UB 1 cosa + UB 2 sina = — 2A sin(7i — a), "n 



UB 1 sina-UB 2 cosa = -2Acos(A— a), J ( 2 °^ 



UB 1 =-2AsinA, UB 2 =2AcosA, (26) 



<p = Vx + A(<f> 1 -<j> 2 ) 



= Ux + A *^(ch ax + sh ax) [cos 6a- cos {ay -f A) + sin bx sin (ay + A)] 

 — Ae^(ch ax — sh ax) [cos 7w cos (ay + h) — sin 6# sin (ay + A)] 



= Vx + C^ 3 cos (ay + A) + Ce^ sin (ay + h) t (27) 



with 



2A = C, UB^-Csinfa UB 2 =CcosA, . . (28) 

 U7? = C( — T/x sin A + t/ 2 cos h), (29) 



TT y-72 



C? dx 2 = (~ Vl C0S 2a + Vi siu 2a ) sin /j + ( Vl sin 2a + ^2 cos 2a) cos A 

 = 77! sin (2a- A) + 7/ 2 cos (2a -A), (30) 



TT 74 



C? a\J = ** sin ( 4a ~ A ) +^2 cos (4a-A), (31) 



From (27) wheny = 0, 



. dd>i A O'^o /-^ r /7 » 



A i^7+ A 2^= Cc[7 7l cos(A-«)+7 ?2 sin(A-a)] (32) 



