334 The Astronomer Royal on the Numerical Expression 



for any number of numerical values of T. To find the effective 

 pressure, the pressure first found must be diminished by the 

 atmospheric pressure, or by the pressure of steam at 100°, and 

 it thus becomes 100 x (K T — K 100 ). The limit of the length of 

 the cylinder will be determined by finding where the steam- 

 pressure = atmospheric pressure. By Fairbairn's formula, 74*638 

 kilogs. of saturated steam at 100° (the quantity which escaped 

 in Mr. Bidden s experiment) occupy 122*28 cubic metres : of 

 this, 0*0746 cubic metre remains in the boiler, taking the place 

 of the water from which it was produced ; the whole volume 

 expelled is therefore 122*21 cubic metres, and the limiting 

 length of the cylinder is 122*21 linear metres. 



13. By these methods Professor Miller calculated the follow- 

 ing corresponding values of z 3 the distance to which the piston 

 has travelled (the unit being the metre), and F the effective 

 pressure on the piston (the unit being the kilogramme). The 

 degrees of temperature are also given, as they are the elements 

 from which z and F are computed ; but they are not in any way 

 used in the subsequent calculations. 



T. 



z. 



P. 



Tees Centigrade. 



Metres. 



Kilogrammes 



152*84 







42185 



150 



1-621 



38356 



145 



4*743 



32162 



140 



9-345 



26615 



135 



14-740 



21668 



130 



21*541 



17271 



125 



30136 



13377 



120 



40*976 



9943 



115 



54-708 



6926 



110 



72*186 



4288 



105 94-325 1991 

 100 122*21 



14. The effective energy of the expanding steam, as shown 

 (for instance) by the momentum communicated to a material 

 piston, will be represented by the integral \ dz . F. As the sym- 

 bolical form of the function F is not known, it is necessary to 

 perform the integration by quadrature. For this purpose I laid 

 down the twelve data of this Table graphically (taking z as the 

 abscissa, and F as the ordinate), and drew a curve by hand 

 through the points so defined. Then I measured the ordinate 

 for each of the values of z ; 0, 1, 2, 3, &c. And I integrated 

 them by the formula \ (first ordinate -f last ordinate) + sum of 

 intermediate ordinates — T ^ sum of second differences ; where it 



