ox THE USE OF LOGARITHMIC COORDINATES. 1G5 



20. The Scale Lines. — A change in o or ft is equivalent to a definite 

 shift of one of the short wave lines, keeping it parallel to itself. This 

 shift might obviously by proper graduations be measured along any line 

 not parallel to the line shifted. By giving the line, along which we wish 

 to measure either of these translations, a proper inclination we can make 

 the lines already on the chart divide this line with the proper logarithmic 

 graduations for a or ft, as the case may be. 



21. Scale Line for a. — This scale line will show how to move the short 

 wave gravity line and the curved portion so as to adapt the chart to new 

 values of gravity, density of liquid and solid, and thickness. The short wave 

 gravity line is the homologue of 



u- 



-a\-. 



If a become na and X become — ^ the value of u is unchanged. 



Therefore a shift to the left of h log n must be made when a becomes n a ; 

 a point on the short wave gravity line must move one large square to the 

 left when the point of intersection on scale moves through two large 

 squares. The scale line will then make an angle tan~i 2 with axis of A. 

 It must be graduated so that its readings increase as we move upwards on 

 the scale line ; the point in which it cuts the short wave gravity line 

 must read the o appropriate to values of g, E, e, s, and p in § 17. But 

 the graduation of the u axis must be utilised to save regraduating the 

 scale line. Thus, look for the value of a on the it line ; or 10±"a where n 

 is a whole number ; a parallel to axis of \ through this point cuts the 

 short wave gravity line in a point through which the scale line may con- 

 veniently be drawn. The graduations are then those of the axis of ?(■ 

 multiplied or divided by some positive integral power of 10. 



The scale line is inserted on the chart and marked ' Scale Line for a.' 

 22. Scale Line for ft. — This scale line shows how to move the short 

 wave elasticity line and the curved portion so as to allow for changes in 

 the values of the density, elasticity, and thickness of the solid. (The short 

 wave elasticity line is not affected by a change in the density of the liquid.) 

 The short wave elasticity line is the homologue of 





If ft becomes nft and \ becomes \/n\, u is unchanged. 



A scale line for ft is inserted on the chart. 



To use the scale lines, move the figure consisting of the two short wave 

 lines and the curve derived from them, without rotation until each of the 

 short wave lines cuts the corresponding scale line in the proper point. The 

 whole operation can easily be accomplished by the use of tracing paper. 

 This translation will obviously move the curve to the correct new 

 position. 



23. Comjylete Curve obtained from Short Wave Curve.— There is now 

 on the chart a curve and two scale lines which enable that curve to be 

 shifted so as to represent 



