On the Strength of Cylindrical Steam Boilers. 69 



separating it along two lines, on opposite sides, would be represented 

 by multiplying the diameter by the force exerted on each unit of sur- 

 face, and this product by the length of the cylinder. But even with- 

 out regarding the length, we may consider the force requisite to rup- 

 ture a single band, in the direction now supposed, and of one linear 

 unit in breadth ; since it obviously makes no difference whether the 

 cylinder be long or short in respect to the ease or difficulty of sepa- 

 rating the sides. The divellent force, in this direction, is therefore 

 truly represented by the diameter multiplied by the pressure per unit 

 of surface. The retaining or quiescent force in the same direction, 

 is only the strength or tenacity of the two opposite sides of the sup- 

 posed band. Here also, at the moment when a rupture is about to 

 occur, the divellent must exactly equal the quiescent force. 



3. In order to nimate the augmentation oi divellent force, conse- 

 quent upon an increase of diameter, we have only to consider that as 

 the diameter is increased, the product of the diameter and the force 

 per unit of surface, is increased in the same ratio. But unless the 

 thickness of the metal be increased, the quiescent force must remain 

 unaltered. The quiescent forces, therefore, continue the same ; the 

 divellent increase witu the diameter. 



4. Again, as rt,he di meter of the cylinder is increased, the area of 

 its end is increased in the*ratio of the square of the diameter. The 

 divellent force is therefore augmented in' this ratio. But the retain- 

 ing force does not, as in the other direction, remain the same, since 

 the circumference of a circle increases in the same ratio as the diam- 

 eter. The quiescent force will consequently be augmented in the 

 simple ratio of the diameter, without any additional thickness of 

 metal, so that on the whole the total tendency to rupture in this di' 

 rection will increase only in the simple ratio of the diameter. 



5. Since we have seen that the tendency to rupture, in both di- 

 rections, increases in the simple direct ratio of the increase of diam- 

 eter, it is obvious that any position of the sheets which is right for 

 one diameter, must be right for all. Hence, there can never be a 

 condition, in regard to mere magnitude, which will require the sheets 

 to be reversed. 



6. The foregoing considerations being once admitted, we may 

 proceed to ascertain what is the true direction of the greatest tena- 

 city in the sheet, if any difference exist, and to what that difference 

 might amount, consistently with equal safety of the boiler in both 

 directions. 



