370 Professor Hamitton on Conjugate Functions, 
a x a X a) is positive or contra-positive ; and on examining the law of its progression, 
(as we lately examined the law of the progression of the square,) we find that the cube 
a x axa increases constantly and continuously with its cube-root a from states inde- 
finitely far from zero, on the contra-positive side, to states indefinitely far advanced 
on the positive side of zero, in the general progression of ratio, so as to pass succes- 
sively but only once through every state of contra-positive or positive ratio, instead of 
first decreasing or retrograding, and afterwards increasing or advancing, like the 
square. Thus every ratio has one and only one cube-root, (commensurable or in- 
commensurable,) although a ratio has sometimes two square-roots and sometimes none, 
according as it is positive or contra-positive ; and when the two extreme effective steps 
a and b’ of the continued analogy (257.) are given, we can always conceive the cube- 
root a of their ratio ~ determined, and hence the two mean steps or mean propor- 
tionals of the analogy, b and v’. 
29. In general, as we conceived a continued analogy or series of equi-distant mo- 
ments, generated from a single standard moment a, by the repetition of a forward 
step a and of a backward step Oa; so we may now conceive, as another sort of conti- 
nued analogy, a series of proportional steps, generated from a single standard (effec- 
tive) step a, by the repetition of the act of multiplication which corresponds to and is 
determined by some one multiplier or ratio a (+E 0), and of the inverse or reciprocal 
act of multiplication determined by the reciprocal multiplier or ratio ua: namely, the 
following series of proportional steps, 
~--Uaxuaxutaxa,taxuda X ay aX a, a,aX a,aA XaAXa,a XAX AX ayaee 
(259.) 
which may also be thus denoted, 
. u (a aa) x a,u(a a) x a,ua@xa,lxa, @x a, @axa, Aaaxa,... (260.) 
and in which we may consider the system or series of ratios or multipliers, 
-» U(a@aa), U(aa), Ua, 1, a, aa, aaa,... (261.) 
to be a system generated from the original ratio or multiplier a, by a system of acts 
of generation having all one common character : as we before considered the system of 
multiple steps (98.), 
-o- Ola 11Ola + Ola, Oal+ Oa, On, 0; a, ata, at ata, 
to be a system of steps generated from the original step a by a system of acts of ge- 
neration to which we gave the common name of acts of multiplying. 
