166 REPORT— 1898. 



for any values of g, p, e, s, and E. The complete curve to represent 



h 



u 



must now be drawn. The above equation can be written 



where a, p, and c have been previously defined. 



When the short wave curve has been placed in its proper position on 

 the chart, calculate c from the equation . 



27res 



From each value of u, as read on the chart, the length 



I [log(\ + c)-logc} 



has to be subtracted. The slide rule is an instrument specially designed 

 for the purpose of adding and subtracting logarithms ; the operation of 

 drawing the complete curve can l^e performed by one setting of a slide 

 rule, without other numerical computation. The top scale on Davis's 

 10-inch slide rule is the same size as the scale of Human's paper ; to find 

 a point on the complete homologue set the right hand 1 on the top scale 

 of the sliding piece under the appropriate value of c on the right hand top 

 scale of the rule ; the rule as thus set is a table of division of all numbers 

 by c. Add any desired value of \ to c ; look for X + c on the top scale 



and take ofi" distance from the middle ' 1 ' of top slide (if is less than 



c 

 10) to X + c on top scale. This length is log (X + c)— log c. Take ofi" this 

 distance on a pair of proportional compasses set to reduce to one-half. 

 This is the length by which the point on the short wave curve must be 

 dropped to become a point on the complete homologue. 



Without resetting the rule repeat the operation with other values of X.i 

 In this way the complete homologue may be drawn without much trouble, 

 and with only so much mental labour as is involved in adding the chosen 

 value of X to the appropriate value of c. 



24. Recapitulation. — We have now upon the chart a curve which is 

 the homologue of the complete equation for waves on a solid covering 



liquid of sufficient depth for coth t^' to be taken equal to unity. The 



X 



case where g, E, e, p, and s have the values given in sect. 1 7 is represented 



on the chart by a thick curved line running beneath the short wave curve. 



The arithmetical operations necessary to get a curve to represent the 



' If the scale of rule have any size other than that of chart the same method may 

 be used by setting the proportional compasses properly. 



The whole operation can be performed without a slide-rule on the chart. Draw 

 a straight line through left hand bottom corner of the paper at an angle tan-' 2 with 

 the horizontal. Use the projection of the left hand coordinates as the one on which 

 to read A + c and c ; use the horizontal scale to read off length i{ log (A + c) -log e] . 



