122 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION 



safety valve for measuring the intensity of pressure, the theory would 

 be incomplete without considering it in connection with the diameters 

 of the pump and cylinder. For which purpose 



Put 3 zz the diameter of the safety valve, expressed in inches or 



parts, 

 and w the weight thereon, or the force that prevents its rising. 



Then, according to the principle announced in Proposition II., we 

 obtain the following analogies, viz. 



D 2 : ^ : : P : w, 

 d* : ^ : : p : w ; 



and from these analogies, by making the products of the extreme 

 terms equal to the products of the means, we get 



D'wzz^P, (93). 



zudd*w = tfp. (94). 



Now, in order to pursue the expansion of these equations, we shall 

 suppose the value of & to be one fourth of an inch, while the numerical 

 values of the other letters remain the same as supposed for the several 

 examples under equation (88) ; then, to determine the corresponding 

 value of w, or the power which prevents the safety valve from rising, 

 when all the parts of the instrument, or the several powers and pres- 

 sures are in a state of equilibrium, we have the following examples to 

 resolve according to the proposed conditions. 



129. EXAMPLE 5. The diameter of the cylinder is 5 inches, that of 

 the indicator or safety valve J of an inch, and the entire pressure 

 upon the piston of the cylinder 18750 Ibs. ; what is the corresponding 

 force preventing the ascent of the safety valve, on the supposition of a 

 perfect equilibrium ? 



Here we have given D zz 5 inches ; 3 zz J of an inch, and P zz 1 8750 

 Ibs. ; consequently, by substitution, the equation (93) becomes 



5 2 wzz.25 2 X 18750; 

 from which, by division, we get 



.0625 X 18750 



wzz zz46.875 Ibs. 



AQ 



But the general expression for the value of iv, as derived from the 

 equation (93), becomes 



_J 2 P 

 - V' (95). 



From which we derive the following rule. 



