252 Researches in the Theory and Calculus of Operations. 



therefore measurable as a cube or magnitude of three di- 

 mensions. 4° The ultimate values of the velocities generated 

 by the forces of the third and second degrees in the time x, 

 are respectively Sx 2 and 2x (differential coefficients of x z and 

 x 2 )=3 and 2 when x=l; and as they are now constant forces, 

 their combined effect in the subsequent unit of time will be 

 equal to their product, Sx 2 x2x=6x 3 , or equal 3.1 2 x2.1=6.1 3 

 in the unit of time immediately succeeding the expiration 

 of the first unit of time occupied in the genesis. The pro- 

 duct of the specific heat by the atomic weight of the chemical 

 elements is equal to the number 6 (Regnault), exceptions 

 probably owing to perturbation. 



"When two constant forces combine their actions, their 

 effect is equal to their product. Gravity is a constant force, 

 and the ultimate value of the velocity generated by a fall- 

 ing body in the time x is 2x, differential coefficient of 

 the integral x 2 ; that is, the differential coefficient is the 

 ultimate value of the force which has generated the in- 

 tegral result while acquiring that ultimate value; and 

 similarly Sx 2 is the ultimate value of the force which 

 has generated the integral x 3 with increasing acceleration. 

 When the atomic weight is 1, the equation Sx 2 x2x=6x 3 is 

 fix 3 



identical Sx 2 = ir -=Sx 2 ; but when the atomic weight is m, 



Zx 



or mx for the future time x (differential coefficient peculiar 



6x 3 6x 2 



to the element), we must have Sx 2 xmx—6x 3 , 3x 2 = — = - 



n 7 mx m 



or 3xm=6 when x=l. The number 3 arises out of the ge- 

 netic process, and expresses the number of linear unit factors 

 requisite to generate or construct a geometrical cube, and 

 m is the reacting force overcome in the operation. The 

 thermometrical measure of 1 degree in rise of temperature, 

 of course denotes the uniform genesis of a unit of volume 

 in a unit of time, beginning with that time at zero. 



The following general principles will be demonstrated in 

 the Third Part: 



