ox THE USE OF LOGARITHMIC COORDINATES. 169 



IV. New values for p. If p changes to np then u and X must become 

 nu and -. 



31. Translation of the Gravity Curve. — New values for e. If e, \. and 

 u are changed to ne, n\, and \/un the gravity equation is unaltered. 

 Thus if e becomes ne move the complete gravity curve log n to the right 



and - log n upwards. 



It is unnecessary to show that similar translations may be found for 

 changes in (/, p, and s. 



32. Scales Lines for Complete Gravity and Elasticity Curves. — Two 

 scale lines would be needed for each curve to permit of complete generality. 

 Thus the gravity curve would have scale lines for changes in y and in 



; the elasticity curve, scale lines for — and — . These scale lines would 

 f P P 



all be put in on the same principles as those in the former part of this 

 paper. 



33. Scale Lines for Thickness. — Two scale lines are inserted on the 

 chart marked ' Elasticity Scale Line for Thickness ' and ' Gravity Scale Line 

 for Thickness.' The first of these lines will be understood at once from 

 paragraph 30, I. 



From paragraph 31 the direction of motion of the gravity curve in order 



to adapt it to new values of e must make an angle tan with the in- 



creasing direction of the ordinate of A. Any line drawn on the chart in 

 this direction can be made use of as a scale line. It is convenient, how- 

 ever, to have the scale line placed in a position so that it cuts the complete 

 gravity curve or its attendant short and long wave gravity lines on the 

 chart. The scale line has been drawn at the appropriate inclination 

 through a point on the short wave gravity line where /\= 10000. Now the 

 thickness of ice is here 100 ; if the scale line be marked 100 at this point 

 it must be graduated in either direction, so that when the curve (and its 

 attendant lines) is moved along the direction of the scale line, the point of 

 intersection of the short wave gravity line and the scale line shall read 

 the thickness on the latter. The scale line must be graduated by a loga- 

 rithmic scale, so that the readings increase ten-fold for every large 

 square moved through to the right : this is done at once by the graduations 

 on the paper without making a special logarithmic scale. 



The complete curve for the case when the values of g, p, s, E are the 

 same as before and e = l is inserted on the chart. It cuts the elasticity 

 scale line for e at graduation 1 . 



If it were only desired to make the chart give at once all values for ic 

 and /\ when one other quantity such as e was varied the second method of 

 treating the original equation would be the better ; but it would become 

 somewhat confusing to use a succession of scale lines in order to adapt the 

 chart to changes in a number of the quantities concerned. If more than 

 one of these has to be altered it would be more convenient to employ the 

 former method. 



It must be noted that the complete curve is calculated from the 

 position of the gravity and elasticity curves by the method of § 11. A 

 small portion only joining the two branches has to be drawn, as the 



& 



