70 On the Strength of Cylindrical Steam Boilers. 
7. Let «=the diameter of the cylinder; f= the force or pressure 
per unit of surface, (pounds per square inch, for example;) T= the. 
tenacity of metal, which with the diameter x and the force f will be 
required in the linear unit of the circumference, in order to hold on 
the head. Then, the whole quiescent force will be 3.14162T, while 
the divellent will be .7854x2f; consequently .785422f=3.14162T, 
as above stated. Dividing by .78542, we have af=4T; and we de- 
aabny : 4T 4T af 
rive immediately se? 2. Je ne 
the longitudinal bar of the assumed unit in width, will be one fourth 
of the product of the diameter into the pressure, measuring the tena- 
city by the same standard as the pressure, whether in pounds or 
kilogrammes. e 
8. Now assuming the tenacity required in the circular band of the 
same width to be t, we shall, agreeably to what has already been said, 
have the divellent force expressed by af and the quiescent by 2t, so 
oe di x ian Ee re 
that 2f=2¢ and t=9 ; also f=—> and «= ry Having thus obtain- 
That is, the tenacity of 
ed two expressions for each of the quantities « and f, we may by 
comparing them, readily discover the relative values of T and ¢; 
4T t 4T 2 
thus, «=— and a= 7 hence te and 4T=2¢ or ¢=2T. 
From which it follows, that, under a known diameter, and witha — 
given force or pressure, the tenacity of metal in a cylindrical boiler — 
of uniform thickness, ought to be twice as great in the direction of 
the curve as in that of the length of the cylinder, and that of this 
could be the case the boiler would still have equal safety in both di- 
rections. In whichever direction, therefore, the rolling of the metal 
‘gives the greatest tenacity, in the same direction must the sheet al- 
ways be bent in forming the convexity of the cylinder. — It follows 
that if we suppose the tenacity precisely equal in both directions, 
the liability to rupture, by a mere internal pressure, ought to be twice 
as great along the longitudinal direction as at the juncture of the 
head. This supposes the strain regular and the riveting not to 
weaken the sheet. ” 
9. To know how large we may safely make a cylindrical boiler, 
having the absolute tenacity of the metal, in the strongest direction, 
and with a known thickness, we have only to revert to the formula 
w=. That is, the diameter will be found by dividing twice the 
