210 
Mr. Davies Gilbert on the 
Then, since the section and the superincumbent pressure 
must always be in the same proportion to each other, 
x and — are in a constant ratio. Let then — — — — 
y my 
where m is the modulus of pressure in the given material ; 
but when x = o, y = a, therefore ~ = the nat. log. — ; 
or ’£—^ i = the tab. log. A = 2,3025851 ; but if e and y 
the homologous sides or diameters of these sections ; then, 
— L— = tab. log. — . 
Finally, I would notice a correction of frequent use in 
practical surveying, to be deduced from the properties of the 
catenary curve. 
When the measuring chain is extended over ground un- 
even, intersected by ditches, or made soft by water, it cannot 
be laid flat, but must be elevated at both its extremities, 
while the middle just touches the surface: thus giving the 
measurement too great by the difference between the whole 
perifery and the double ordinate. 
Let % = the half length of the chain. 
x = the elevation at each end equal to the depths of 
curvature. 
Then Equ. B. No. 2. y = a x nat. log. 
And Equ. A. No. 3. a = - — — ; therefore 
2 X 
y 
Z — X 
2 X 
1 Z + X 
x nat. log. 
But when x is very small in comparison of z, the nat. log. 
of 
Z + X 1 21 , 
becomes — , and 
Z — X z 
y 
X 1 - 2 X 
X 
z X z 
~ z 
X ‘ 
z 
is therefore the difference between half the chain and the 
