THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
407 
The Case 52, as belonging to a different series of Problems. Art. Nos. 22 to 36. 
22. In the foregoing Case 52, where all the curves are one and the same curve, we 
have the unclosed trilateral aBcDeFg, and we seek for the number of the united points 
(a, g). But we may consider this as belonging to a series of questions, viz. we may 
seek for the number of the united points (a, B), (a, c ), (a, I)), (a, e), (a, F), (the last 
four of these giving by reciprocity the numbers of the united points (B, D), (B, e), 
(B, F), (B, g)), and finally the number of the united points (a, g). It is very instructive 
to consider this series of questions, and the more so that in those which precede (a, F) 
there are only special solutions having reference to the singular points and tangents of 
the curve, and that the solutions thus explain themselves. 
23. Thus the first case is that of the united points ( a , B), viz. we have here a point 
a on the curve, and from it we draw to the curve a tangent «B touching it at B ; the points 
a and B are to coincide together. Observe that from a point in general a of the curve 
we have X— 2 tangents (X the class as heretofore), viz. we disregard altogether the 
tangent at the point, counting as 2 of the X tangents from a point not on the curve, 
and attend exclusively to the X — 2 tangents from the point. Now if the point a is an 
inflection, or if it is a cusp, there are only X — 3 tangents, or, to speak more accurately, 
one of the X— 2 tangents has come to coincide with the tangent at the point; such 
tangent is a tangent of three-pointic intersection, viz. we have the point a and the point 
B (counting, as a point of contact, twice) all three coinciding ; that is, we have a position 
of the united point ( a , B) ; and the number of these united points is 
24. It is important to notice that neither a point of contact of a double tangent, nor a 
double point, is a united point. In the case of the point of contact of a double tangent, 
one of the tangents from the point coincides with the double tangent ; but the point B 
is here the other point of contact of this tangent, so that the points a , B are not coin- 
cident. In the case of a double point, regarding the assumed position of a at the double 
point as belonging to one of the two branches, then of the X — 2 tangents there are two, 
each coinciding with the tangent to the other branch ; hence, attending to either of 
these, the point B belongs to the other branch, and thus, though a and B are each of 
them at the double point, the two do not constitute a united point. (In illustration 
remark that for a unicursal curve, the position of a answers to a value =X, and that of 
B to a value of the parameter d, viz. X, g> are the two values of 0 at the double 
point; contrariwise in the foregoing case of a cusp, where there is a single value X=gj. 
Hence the whole number of the united points ( a , B) is and this is in fact the 
value given (as will presently appear) by the theory of correspondence. 
I recall that I use A, =2D, to denote twice the deficiency of the curve, viz. that we 
have A=X— 2tf+2+;c, = _2a?— 2X+2+g. 
25. The several cases are 
United points. 
(a, B) b-f3-f3'=2A, 
(«> O c -y -y' +2(b — /3— (3')=(X-2)A, 
