I 



ON THE USE OF LOGARITHMIC COORDINATES. 163 



where li, A, g have their usual significance, and 



e= thickness of the solid assumed uniform 

 E=Young's modulus for the solid 

 p^ density of the liquid 



s= density (not superficial density) of the ice 

 /(.^uniform depth of the liquid, 



all expressed in some consistent system of units throughout this discus- 

 sion. 



16. Fundamental Assumption. — Let us assume that h is large enough 



compared with -- for us to write 



coth- — = 1 

 \ 



without appreciable error. The equation now becomes 



^ IttVE 



> 



which is of the form 



1+^ 



u-= . 



1+- 



This is a non-translatant ; if we regard \ and u as the independent and 

 dependent variables, and change a, h, and c into a', b', and c', it is impos- 

 sible to find two numbers n and vi such that the equation 



a'mX + 



1+^ 



ni\ 

 is identical with the preceding. 



Construction op a Chart for Waves on a Frozen Sea. 



17. Short Wave Gravity and Elasticity Lines. — When X is sufficiently 



small for unity to be negligible compared with - the equation becomes 



A, 



where 



and 



n'^=aX'^ + ^, 



a = -1^-, 

 47r2es' 



^ ^ TT^e^E 



39 



Mi' 



