ee 
3 On the Calculus of Operations. 329 
how, when the primitive function is given, al] the suecessive de- - 
rivatives are to be obtained from it; but the theory of generation 
evolves the primitive, together with its derivatives, all by one 
same process. 
Generation is still an operation, but characteristically distin- 
guished by its result, which is, in general, itself a generator, or 
an operutor of an order or rank next inferior to that of its own 
generator. : 
A constant velocity causes a body, or material unit, to describe 
a distance which increases uniformly, or as the time; and will 
therefore generate the distance unity (linear unit) in the unit of 
time, and consequently the distance whose ratio is x in @ units of 
time. A constant velocity, therefore, has but one measure of 
Operation, namely the unit of distance, and thus corresponds ex- 
actly with multiplication by linear unity. 
_ A constant force generates velocity uniformly ; and this veloc- 
ity, from the very commencement of its existence, operates upon. 
the material unit, and causes it to describe the unit of distance 
in the first unit of time. But at the expiration of this first unit 
of time, the amount of velocity, having increased uniformly dur- 
ing that time, is evidently such as will of itself cause the mate- 
rial unit to describe iwice the distance.in the immediately suc- 
interval occupied in the 
j ich the result itself is 
velocity increases as the duplic 
as the square of the time; and we have now discoveted a method 
which enables us to raise the dimensions of a ratio £ 
Assume a series of subordinated generating forces, or more 
generally a hierarchy of generating powers, such that the power 
of the lowest order shall correspond to a constant velocity, which 
generates distance (power of the order zero) uniformly ; the 
power of the second order shall correspond to a constant force, 
which generates velocity uniformly ; the power of the third or- 
"Stoop Seams, Vol. XI, No. $3 —May, 1851. 42 : 
