304 Professor Haminton on Conjugate Functions, 
because now we either do not alter the means B and c at all, or else alter them oppo- 
sitely in direction and equally in degree. And similarly, 
Rip ee Dy SSCS 3h 
and Dp’ =peha ah (24:.) 
then p’'—c=B—a’, 
because here we only alter equally, if at all, in degree, and oppositely in direction, 
the extremes, a and p, of the first analogy. It is still more evident that if two pairs 
be analogous to the same third pair, they are analogous to each other ; that is 
fee — co —seAe 
and B—A=D—C, (25.) 
then D—c=p'—c. 
And each of the foregoing conclusions will still be true, if we change the first supposed 
analogy p—c=B—A, to anon-analogy of subsequence D — c> B — A, or to anon-analogy 
of precedence p —c <B—A, provided that we change, in like manner, the last or con- 
cluded analogy to a non-analogy of subsequence in the one case, or of precedence 
in the other. 
It is easy also to see, that if we still suppose the first analogy p—c=B—A to 
remain, we cannot conclude the third analogy, and are not even at liberty to suppose 
that it exists, in any one of the foregoing combinations, unless we suppose the second 
also to remain: that is, if two analogies have the same antecedents, they must have 
analogous consequents ; if the consequents be the same in two analogies, the antece- 
dents must themselves form two analogous pairs ; if the extremes of one analogy be 
the same with the extremes of another, the means of either may be combined as 
extremes with the means of the other as means, to form a new analogy ; if the means 
of one analogy be the same with the means of another, then the extremes of either may 
be combined as means with the extremes of the other as extremes, and the resulting 
analogy will be true ; from which the principle of inversion enables us farther to 
infer, that if the extremes of one analogy be the same with the means of another, 
then the means of the former may be combined as means with the extremes of the 
latter as extremes, and will thus generate another true analogy. 
On continued Analogies, or Equidistant Series of Moments, 
5, It is clear from the foregoing remarks, that in any analogy 
B'— A’ =B—A, (26.) 
the two moments of either pair A B or A’ B’ cannot coincide, and so reduce themselves 
P ’ 
