168 Lord Ilayleigh on the Nodal Lines of a Square Plate. 



equal to unity. The intervals were then bisected by means of 

 the appropriate interpolation-formulae involving the use of four 

 terms. If p, q, r, s be four consecutive values of z 3 the interval 

 between q and r is bisected by 



... (3) 



q + r q + r—(p + s) 



2 



16 



At the ends of the Table a modification is required. For the 

 term next before that corresponding to the middle of the rod we 

 have s=q, as we know that every thing is symmetrical about the 

 centre. For the term corresponding to #=O025, 1 have con- 

 tented myself with a simple interpolation by first differences, 

 a course justified by the fact that in the neighbourhood of the 

 ends the curvature is exceedingly small. The Table stands as 

 follows : — 



X. 



z. 



Diff. 



X. 



z. 



Diff. 



•000 



-1-6448 





•250 



+ •1632 



1577 



•025 



1-4538 



•1910 



-275 



•3110 



•1478 



•050 



1-2628 



•1910 



•300 



•4475 



•1365 



•075 



10724 



•1904 



•325 



•5713 



•1238 



•100 



•8835 



•1889 



•350 



•6814 



•1101 



•125 



•6966 



•1869 



•375 



•7765 



•0951 



•150 



•5131 



•1835 



•400 



•8559 



•0794 



•175 



•3340 



1791 



•425 



•9183 



•0624 



•200 



- -1607 



•1733 



•450 



•9636 



0453 



•225 



-j- -0055 



•1662 



•475 



•9908 



•0272 









•500 



1-0000 



•0092 



It appears that the displacement at the ends of the rod is by 

 no means of the same numerical magnitude as at the middle. 

 It will be found by interpolation that r = — 1 when ^ = '0846, 

 which last number therefore gives the nearest approach made by 

 the nodal line of the resultant vibration to the sides of the 

 square. The nodal lines of the original systems correspond to 

 # = •2.242, differing sensibly from \. 



If we take two adjacent sides of the square as axes of Carte- 

 sian coordinates, the lines of constant displacement are repre- 

 sented by 



z x -\-z y - .const., (4) 



z being the function already investigated and tabulated. For 

 the nodal line the constant in (4) is to be put equal to zero. 



In order to construct (4) graphically, the most convenient 

 method is that adopted by Maxwell in similar cases. The sys- 

 tems of curves (in this case straight lines) represented by 

 z x — const, and z y — const, respectively are first laid down, the 

 values of the constants forming an arithmetical progression with 

 the same common difference in the two cases. In this way a 



