Researches in the Theory artd Calculus of Operations. 253 



1° Beginning at zero, when a generating force is constant, 

 its generated effect increases uniformly with the time oc- 

 cupied, a unit of increment is added in each successive unit 

 of time, becoming x for the time x. If the sum of these in- 

 crements is itself a generating force, its (ultimate) value 

 will be 2x, when measured by the effect it will generate as 

 a constant force in the equal subsequent time x. 



2° When the generating force increases uniformly, its 

 effect is accelerated, and increases as x 2 the square of the time, 

 the ultimate value becoming 3x 2 when measured by its effect 

 in the subsequent equal time x. 



3° When a generated force increases as the square of the 

 time, and is itself a generator, its effect increases as the cube 

 of the time, etc. 



4° Each generated force is one degree lower in intension 

 than its generator; so that a force of the first degree will 

 be reached, whose effect in the time x is a force of the 

 degree zero, expressed in a power of x one degree higher 

 than that of its generator. 



Writing out the successive terms of the genesis, begin- 

 ning with a constant force of the third degree, for two succes- 

 sive intervals of time x (past) and h (future), we get the 

 following tablet: 



x. xl t . hl t . 



1<£'": 







1.2.3 

 1.2 



x$": 



2xcp", 



-^-xhcp , 



2.3.1 

 1.2 





Sx 2 (p', 



^X 2 h(jf. 





x 3 <p°. 



lx 3 <p°. 







xh 2 (p°. 



The lower diagonal line is the third power of (z-f A). 



The force of heat, when employed to produce motion 

 through the conversion of water into steam, may be inter- 



