ON THE USE OF LOGARITHMIC COORDINATES. 171 



chart, and also the minimum value of u is obtained at once by inspection 



without the trouble of substituting X=419 in the original equation. 



Logarithmic coordinates may be of use to find the roots of equations of 



high orders. 



37. The Homologues of Equations representing the Relation of u to\ 



e 

 en -=som( 

 \ 



tion becomes 



when ~:=some constant — In the original equation let (=1* The equa- 



9^ 



2- 



^^^ 



9^ 



For all values of \ included in the chart ^ may be neglected with 



respect to S • the homologue is a straight line identical with the elas- 



P 

 ticity scale line for thickness (if we take E, n, s as before). In the same 



way horizontal lines drawn through the points of intersection of lines A. 

 =2, X=3, <tc. with the elasticity curve for ice 1 unit thick give the velocity 



of propagation of waves on ice which is always ^, ^j '^^- thick. The size of 



the waves makes no difference so long as we confine ourselves within themaxi- 

 mum range of X on the chart. This process cannot, however, be continued 



indefinitely ; even when e becomes - the line becomes slightly bent 



upwards as A approaches 10^ ; when «:=-—_- or less the relation is 



expressed by the long wave gravity line already on the chart. 



A horizontal line ■?; = 245000 gives the velocity of propagation of 

 waves when e=A/l-61 ; no matter what E may be, the velocity of waves 

 whose length is 1*61 times the thickness of the ice (when p=s=\) is 

 equal to the velocity of propagation of sound in the ice. The line is 

 marked ' Sound Line ' on the chart. 



The Use of Losahithmic Cooedinates to pixd an Approximate Equation 



CONNECTING A SeEIES OF EXPEEIMENTAL EeSULTS. 



38. In practical science this arithmetical problem often arises. If the 

 numbers are merely to be recorded graphically it is always better to plot 

 them logarithmically, as the sensitiveness of the record is independent of 

 the distance of a point from the origin. 



If the law which governs the numbers is a straight line law, as is often 

 the case, of course ordinary section paper should be employed. But if on 

 plotting the curve on squared paper it seems to follow some other law, 

 recourse should be had to logarithmic plotting. If the observer com- 

 mences by trying to find an equation of the form 



by arithmetic he will probably restrict his trial to simple values of a and 

 k. By the use of logarithmic paper he avoids the trouble of looking up 

 the logarithms (as Herschel had to do) and gains in accuracy. 



