36 A SYSTEM OF ACCURATE MEASUREMENT 
w= weight of 1 chain of riband. 
, 7” =tensions at any points in curve. 
7=mean of any two values of 7’= ote 
t= standard tension of riband. 
p=radius of curvature at any point of curve. 
G= ss measure of any angle. 
£,¢ =any angles between axis y and any chord of the curve. 
The Pridamental equations to the catenary curve are : 
‘<5 ae :) a hs See ) 
Pw 
IT1.—£quation h? - th - a = Q, 
Uf -1 
x=., therefore L=c (x 2) . By exponential theorem, 
ae 
l 
5 
e gi ak ys e 
es 2 herefore L=/+ 
~ 96" BA —T8e * see B5406 Eh See 
? ig Pu Put 
Met To20e+ & Gao therefore epee es &e, 
PB 2 
But Z-l=(h- = very approximately, therefore (h - ieee sik 
about, the second and omitted term on the right-hand side of 
this equation being very small. The effect of this omission is 
moreover partly neutralized by the omission of a small quantity 
which should be included in the left-hand side. Dividing each 
lk wu 
side a —and multiplying by h? gives h* — th? — ad =0. 
For a “bispik of 10 chains, weighing in all 10 lb, and standard 
len at a tenison * Se i Ib, the result given by the above 
equation is A = 93° 94 lb. The true tangential tension to 
y9 
It has been stated that the curve may be taken as an arc o 
a circle without aad error. The following solution will 
e this manifes 
= Des =¢, very approximately, = id 
z= Gand = sin 6. 
Expanding, 6= OMe wo Se ae + te 
- Maliplying by p= net sont se 
