162 THE STRUT PROBLEM. 
is sufficient to ensure some variation from the ideal 
condition, and, as the deflection is nearly infinite in its 
ratio to eccentricity of loading when the load is nearly 
the Euler load (see equation (18) ), an infinitely small 
eccentricity may cause a definite deflection; in the 
language of mathematics, 00 x O = a; more strictly— 
nearly «¢ xX nearly O = nearly a. 
It is inconceivable that an absolutely different set of 
conditions obtains in the two cases of (a) absolute centrality, 
and (6b) infinitely close to absolute centrality, yet this is 
what Smith apparently assumed.* 
The explanation of the apparent anomalies lies probably 
in the fact that the primary differential equation solved 
by Euler, and later in a modified form, by Smith, is not 
itself exact. This important point seems to have been 
missed in the numerous discussions of column formulae. 
Let H be the load applied. 
The exact equation is 
d2y dy \243 
df fap no T {i+ alt } Lew, Lateonate (14) 
dx? dx 
d?y 
the ordinary equation assumed, see (1), is = — Hy 
dx? 
dy \2 
that is (2 , being small, is neglected, yet it is quite 
dx 
conceivable that even though negligible so far as arith- 
metical results are concerned, yet if taken into account, 
it may provide the element of stability. 
* “The error is not in pure mathematics. From first to last 
Grashof’s careful and elaborate investigation is correct, so far at least 
as I have detected. His mathematical deduction from his final equation 
is substantially right, but his mistake consisted in assuming that the case 
e=O was one which commonly occurred in practice, and thus in inferring 
that the mathematical results of assumption has a bearing on the practical 
question of the strength of struts. This case never occurs in practice, 
and although e may often be very small, still its slightest variation from 
absolutely O altogether destroys the validity of the conclusions drawn.’® 
(Extract from Professor Smith’s Paper, 1878). 
