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ON THE PROBLEM OF THE IN-AND-CIRCÜMSCRIBED TRIANGLE. [514 
17. But this cannot happen if the curve a is distinct also from the curves c, D; 
or, say, simply when a is a distinct curve. The conclusion is, that in the case where 
a is a distinct curve we have 
g = X + X - Red -> 
where the term “ Red.” vanishes except in the case of the identity B = F of the 
curves B, F; and that when this identity subsists it is = a times the reduction in 
the order of the locus of a considered as a free angle; viz. this consists of a first- 
mode and a second-mode reduction as above explained. 
Article Nos. 18 to 21. Remarks in regard to the Solutions for the 52 Cases. 
18. Before going further I remark that the principle of correspondence applies to 
corresponding and united tangents in like manner as to corresponding and united 
points, and that all the investigations in regard to the in-and-circumscribed triangle 
might thus be presented in the reciprocal form, where, instead of points and lines, 
we have lines and points respectively. But there is no occasion to employ any such 
reciprocal process; the result to which it would lead is the reciprocal of a result 
given by the original process, and as such it can always be obtained by reciprocation 
of the original result, without any performance of the reciprocal process. 
19. It is hardly necessary to remark that although reciprocal results would, by 
the employment of the two processes respectively, be obtained in a precisely similar 
manner, yet that this is not so when only one of the reciprocal processes is made 
use of; so that, using one process only, it may be and in general is easier and more 
convenient to obtain directly one than the other of two reciprocal results; for instance, 
to consider the case B = D = F rather than a = c = e, or vice versa; and that it is 
sufficient to do this, and having obtained the one result, directly to deduce from it 
the other by reciprocity; but that it may nevertheless be interesting to obtain each 
of the two results directly. 
20. It is moreover obvious that although the several forms of the same case, for 
instance Case 2, a = c, a = e, or c = e, are absolutely equivalent to each other, yet that, 
when as above we select a vertex a, and seek for the number of the united points 
(a, g), the process of obtaining the result will be altogether different according to the 
different form which we employ. For instance, in the case just referred to, if the 
form is taken to be a = c or c = e, then the equation g = % + x applicable to it; 
but not so if the form is taken to be a = e. It would be by no means uninteresting 
in every case to consider the several forms successively and get out the result from 
each of them; I shall not, however, do this, but only consider two or more forms of 
the same case when for comparison, illustration, verification, or otherwise it appears 
proper so to do. The translation of a result, for instance, of a form a — e or c = e 
into that for the form a = c = x is so easy and obvious, that it is not even necessary 
formally to make it.