254 Researches in the Theory*and Calculus of Operations. 



preted as a force of the third degree. The second line of 

 the tablet applies to this case. Keeping the furnace con- 

 stantly and uniformly fed with fuel, it will send to the boiler 

 a uniform supply of heat, expressed by x in the time x, the 

 temperature beginning at 0° centigrade. Every successive 

 increment of heat is destroyed in its turn by increasing the 

 extent of vibration of the atoms of water from the state a 

 (fig. 7), until finally reaching the state b at the expiration 

 of the time x; but all the destroyed forces are successively 

 restored by the immanent force of the water, so that there 

 is finally an accumulated amount equal to 2x at the end of 

 the time x, during which time the temperature has risen 

 from 0° to x°, and steam is formed. The fire remaining con- 

 stant, the water now receives a constant accession of x° of 

 heat in the time x, so that it is now uniformly increasing 

 force (a force of the second degree), and therefore generates 

 the amount 2.3£ 2 of steam force in the time x; which amount 

 of expansion is expended in overcoming the resistance of 

 loaded machinery, is then condensed to liquidity and re- 

 turned to the water in the boiler, whence an equal amount 

 of steam is advanced in every succeeding time x, and the 

 result is expressed by 2.3x 3 expended in the time x in over- 

 coming the friction of the burthen. 



Making x=h—l, it is seen that x generates the cube x 3 or 

 l 3 in the first interval xor 1; that 2x, the ultimate value of 

 x, generates 2.3x 2 (twice the ultimate value of x 2 ) or 2.3.1 2 

 in the second interval x or 1 ; and that 2.3:r 2 generates 2.3.x 3 

 or 6.1 3 in the third interval x or 1, or 3a; 3 or 3.1 3 while ac- 

 quiring its ultimate value in the second interval x or 1. 



Difference in strength or intensity of the equilibrating 

 calorific forces distinguish the different material elements 

 in respect to their individual density, etc. : the greater the 

 intensity of two equal opposing forces, the tighter they hug 

 together, and consequently the smaller the amplitude of 

 excursion. The three different states of matter, solid, fluid, 



