Skating on Thin Ice. 13 



of a train of artillery, with the ice in waves advancing 

 underneath at the same rate. 



Or else reduce to a steady motion by giving the ice and 

 water underneath a reversed velocity U, with the logs now 



stationary, and -j' replaced by U 2 ~r\. 



Then, with no pack-thrust P, 



?—*£- » 



„ T #i? XJ 2 d 2 v u 2 -U 2 n ,„, 



and with E =pk, I=-T2* 3 5 <r=pe, 



1 . 8 ^ , U*<*V , ^ 2 -U 2 n ... 



5^ +# 7a? +v+ -^- = a -' • • (4) 



Measuring y and 77 upward, the velocity function 



^ = Ua? + A 1 ^i + A 2 2 , ....,, (5) 



= _# = _XJ— A ~— A -^ 



(6) 



2g g \ 1 dx 2 dx /' 



to the first order, where fa <£ 2 are composed of terms of the 

 form g<° +w K*+y») • and taking the water deep, so that fa (f> 2 

 are zero for y— — co , we put 



1 = ^+* cos ( ay _^ + /li ^ $ 2 = e- ax+by cos(ay + lx + h 2 ), (7) 



for each stretch of water between two logs, with some slight 

 discontinuity at a vertical plane of junction, of vertical 

 slipping without cavitation, which may be ignored ; and 



then v 2 (^ 1; <W = o, 



satisfying the Equation of Continuity. 



With the origin midway between two logs, take 



v = B lVl + B aVa , (8) 



7] 1 = ch ax cos bx, rj 2 = sh ax sin bx, . . (9) 

 P 7) d = sh ax cos bx, 7]^= chaxsinbx. . . (10) 



