THE IN-AND-CIECUMSCEIBED TRIANGLE. 
385 
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=%+%' is 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 com- 
parison, illustration, verification, or otherwise it appears proper so to do. The transla- 
tion of a result, for instance, of a form a—e or c—e into that for the form ci—c—x is so 
easy and obvious, that it is not even necessary formally to make it. 
21. I do not at present further consider the general theory, but proceed to consider 
in order the 52 cases, interpolating in regard to the general theory such further discus- 
sion or explanation as may appear necessary. In the several instances in which the 
equation g=x-Fx! i s applicable, it is sufficient to write down the values of f, the 
mode of obtaining these being already explained. 
The 52 cases for the in-and-circumscrihed triangles. 
Case 1. No identities. 
p/=BcD<?F«, f='Fe'DcBa(=x), 
g=2«ceBDF. 
Case 2. a—c=x. 
X=B(x— l)T)eFx, f~YeBxYfx— 1 )(=%), 
g = 2x(x— l)cBDF. 
Second process, for form a=e=x. The equation of correspondence is here 
g-x— x'+ F ( e ~ z— ; 
but the points e being given as all the intersections of the curve «(=<?) by the line-system 
cDe which does not pass through a, we have e — s — s’=0; so that g=x,Tx' > and then 
X=BcDxF(x- 1), — F(a? — l)T)cBx, 
giving the former result*. 
* Of course, the result is obtained in the form belonging to the new form of specification, viz. here it is 
=2x(x— l)cBDF ; and so in other instances; but it is unnecessary to refer to this change. 
MDCCCLXXI. 3 H 
