ON THE USE OF LOGARITHMIC COORDINATES. 107 



equation for any values of the quantities involved consist merely in the 

 computation of o, /3, and c. 



Thus it is possible to obtain by means of logarithmic coordinates a 

 curve "iving ^o for any value of A, without a single operation of the nature 

 of substituting arbitrary values for a variable in the equation. 



If it had been merely desired to draw such a curve the task might be 

 regarded as completed ; but the equation under discussion may be made to 

 yield many interesting illustrations of the method of homologues. Some 

 of these will now be dealt with. 



Another Method of Treating the same Non translatant. 

 25. Long Wave Elasticity Line.—li the motion were controlled by 

 elasticity alone the equation would be 



h 



and if X were so large that ~ could be neglected with respect to unity, the 

 above becomes 



The straight line on the chart mai'ked ' Long Wave Elasticity Line ' is 

 the homologue of this equation. It makes an angle tan - with the 



decreasing direction of the axis of X, and in the diagram is placed for 

 the case when e, E, and ^ have the values above quoted. 



26. — Short Wave Elasticity Line.— This is the homologue of «'-=-— ^ 



and is already on the chart, for the case when the quantities concerned 

 have the values already taken. 



27 .—Cornplete Elasticity Curve.— li the waves were governed by 

 elasticity alone, the homologue showing the value of u for a given value 

 of X would be coincident with the short wave elasticity line when X was 

 small, with the long wave elasticity line when X was great, and would run 

 beneath both lines and leave them to the greatest extent where they 

 cross. To draw the complete elasticity curve we have only to multiply 



each « by ^ / ^ , which can be done by one setting of the slide rule as 



V X-f-c 

 previously shown. 



28. Tlie Two Gravity Lines and Complete Gravity Curve.— By exactly 

 similar methods the two gravity lines and the complete gravity curve can 

 be inserted on the chart. The long wave gravity line is the homologue of 



u~^a\., 

 The short wave gravity equation is 



%''=- X-, 



C 



and the complete gravity curve is got by mechanical computation, by the 



