BEAM OP EECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OE LOAD. 115 
sechand all its differential coefficients will be-of order when z is lar^e. 
Similarly sech® — and its differential coefficients will be of order when 2; is large. 
We see, therefore, that the first 11 terms of the series for I will be of the form 
(algebraic polynomial of degree n in z) e~2 to the first approximation when z is large. 
Further we have obtained an expression for the remainder which is small 
independently of z, for any given large value of n. We see therefore that, n being 
assigned, we may make z as large as we j^lease and I will eventually tend to zero, 
gm/2 t)00oming large more rapidly than any polynomial of finite degree, if z be large 
enough. 
Now z = xlh. We see therefore that, if h be small, the pressure, after a certain 
value of X, decreases with extreme rapidity as we get away from the neighbourhood 
rw 
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1 
1 
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Fig. iv. 
of the concentrated load, because, 2 being then large, even for moderate values of x 
the influence of the exponential term will be predominant. On the other hand, if h 
becomes fiuite, or even large, the algebraic polynomial factor will become predominant, 
and the decrease as we go away from the point of loading will become much less 
rapid. The expansion ( 93 ) gives us a link, as it were, between the case of a very 
thin beam, where the local efiects die out according to a negative exponential of the 
distance along the axis, and that of an infinite solid, where they decrease as an 
inverse power of the distance from the point of loading. 
A diagram is given in fig. iv. showing the variation of the pressure Q along the 
Q 2 
