586 Prof. E. H. Barton on the Lateral 



and enable ns by equating them to find m. Thus, omitting a 

 constant multiplier, we may write, from the second form of (18), 



u = (cos m — cosh m) (cos oc ' -f cosh x f ) 



+ (sin m -f sinh m) (sin x' + sinh x') , . . (19) 



or a corresponding and equivalent expression from the first 

 form of the equation. 



Series of Tones. — We further derive from the two equations 

 (18) the following relation 



cosh?» = l,^ 

 = cosh?ft. J 



cos m 

 or ... . (20, 



sec m 



This may be solved by the aid of tables of hyperbolic 

 cosines, or by a method of expansions and successive approxi- 

 mations as adopted by Lord Rayleigh. 



It may also be solved graphically, as shown in the upper part 

 of fig. 4 (PI. XIV.), the method "being as follows. Take the 

 second form of equation (20) and plot the graphs y = sec x 

 and ?/ = cosh x. Then the values of x at their intersections 

 will give the roots of m sought. 



It will be seen from the upper half of the figure that, 

 apart from the first value which is zero, the series approximates 

 to 3, 5, 7 . . . times 7r/2, the approximation becoming closer as 

 Ave proceed to the higher values. Now the frequencies, as 

 seen from equation (9), are proportional to m 2 ; hence for the 

 higher values they approximate to the squares of the odd 

 numbers. But, since the lowest frequency or prime tone 

 departs distinctly from this simple approximation, the entire 

 series of tones, when expressed as to their relation to the 

 prime, seems disturbed and complicated. Hence the ad- 

 vantage of the graphical view of the matter which exhibits 

 the essential simplicity of the whole. 



The lower part of the diagram (fig. 4) refers to another 

 equation for the fixed-free bar to be dealt with presently. 



Table II. gives the values of m and the relation of the 

 series of tones for a free-free bar, and, as will be shown later, 

 these apply also to a fixed-fixed bar. 



The first column of Table II. is from Rayleigh's values, 

 obtained by computation. The second column follows at 

 once, while the third is derived by logarithmic methods and 

 the fourth expresses the same results in musical notation, the 

 accents representing higher octaves, but the letters do not 

 denote any absolute pitch but are relative only. 



