390 Proceedings of the Royal Society of Edinburgh. [Sess. 
XXVI. — Comparison of Mr Crawford’s Measurements of the 
Deflection of a Clamped Square Plate with Ritz’s Solution. 
By Professor C. G. Knott (Secretary). 
(MS. received July 24, 1912.) 
When Mr Crawford’s paper (see pp. 348-389) was passing through the 
press, Professor Love drew our attention to a paper by Walther Ritz, 
whose early death robbed the world of science of one whose brilliancy 
gave promise of a great future. This paper was published in Crelles 
Journal in 1908 under the title, “Ueber eine neue Methode zur Losung; 
gewisser Yariationsprobleme der mathematischen Physik.” In illustration 
of this new method Ritz showed how to solve, by a series of successive 
approximations, the problem of the clamped rectangular plate. 
I propose here to work out to the third approximation the numerical 
details of the deflections of a square plate subjected to a constant pressure 
on the one side. 
Let z be the displacement of the point x, y, in the originally plane 
square plate of side a , referred to one corner as origin, — then Ritz’s 
solution is, to the third approximation, 
f = 0 - 6744 % + 0 - 0308 ( 4 % + 4 %) + 0 - 00324 % 
+ O-OOKk^ + £ 5 ^) + 0-0004(£ 3 77 5 + £ 5 t 7 3 ) + O'OOOO^ , 
where 10Y/8a 4 is proportional to the pressure, which is supposed to act 
uniformly on the one side of the square plate ; where 
c ELx hL n x ( . K v x . , K+c\cos EL - cosh EL 
$ n = cos " - cosh — n — sin — — - sinh ) — ? , 
a a \ a a /sin K n - sinh K n 
77 , j = similar function of y, 
and where K w is the nth root of the equation : 
cos K cosh K = 1. 
Only the odd values of n enter into the expression for z ; and for 
present purposes it will suffice to write down the values of K v K 3 , K5. 
These are: 
K 1 = 3tt/2 + 0-01765= 4-7300 
K 3 = 7tt/ 2 + 0-00003 = 10-9956 
K 5 = ll7r/2+ = 17-2787 
