12 Sir G. Greenhill on 



side of 0, and with a change to measuring- nj downward, 



9; = A ch j« cos g'^ + B sh qx sin ^ ; . . (2) 



and to satisfy the condition -—^ =0 when #= + Z, 



9; = a[ch$ f (Z + ^?) sin<7(Z — x) + chq(l — x) sing(Z + #) 



+ shg'(Z + #) cos^(Z-A') + sh2(Z— x) cosq^ + x)"], . (3) 



— ^= 2#a[sh<7(Z + #) sing(Z — ^) — sha(Z — x) sin£(Z + #)], (4) 



with an area for buoyancy, 



W C l a 



— = I rjdx = 2- (ch 2gZ — cos2aZ). . . (5) 



When 



x = 0, 770 = 2a (ch #Z sin gZ -f sh ql cos #Z) , 



x=+l, 7j l = 2a (ch ql sh gZ + cos </Z sin gZ) , 



and putting W = 2Zw, so that w is the average superficial 

 loading, which we may reckon in lb/inch 2 , 



_ 2w ql(ch ql sin ql + sh ql cos ql) , fi x 



^°~ p ch 2?Z - cos 2?Z , • • •• W 



_ 2w ql(ch ql sh ql + cos #Z sin ql) ,_. 



^~7" ch 2ql - cos 2ql '' ' * ^ > 



reducing in each case when ql = to 



Vo = Vi = W/» = W (8) 



the uniform depression of the ice when the load is distributed 

 uniformly. 



Equation (8) (Hty § 10 of the B.M. at 0, and skin stress, 

 is changed to a maximum 



E i e z 2o*B- - W W ch gZsin g Z-sh g /cos?Z _1 2 . . 

 *X1 elq *- 6 keq W ch 2ql- cos %ql " 6^ ' W 



- = 2^a 2 (ch^Z singZ — sh^Zcos^Z). . . . (10) 

 a 



In this way the relation can be calculated between the 

 thickness of the ice and the loading permissible, distributed 

 as infantry or artillery. 



13. Next, roll the logs in a regular procession at velocity U 

 and equal interval 21, to represent the passage for instance 



