THE IN-AND-CIliCUMSCEIBED TRIANGLE 
381 
tion with the curve a determine the points g. Considering these as the points (P, P') 
of the general theorem we have j» = l : I change the notation, and instead of a — a — a! 
write g— % — f ; viz. I take (g) for the number of the united points ( a , g ), and 
suppose that the points ( a , g) have a (%, f) correspondence. The most simple case is 
when the curve a is distinct from each of the curves e, F ; here all the intersections of the 
line-system eF g with the curve a are points g , that is we have only the correspondence 
( a , g) ; and since the line-system eF g does not pass through the point a , we have simply 
g— %-x — o. 
5. But suppose that the curves a , e, F are one and the same curve, say that a—e—F ; 
understanding by the point F the point of contact of a line eF g with the curve a , then 
the intersections of the line-system eF g with the curve a are the points g each once, the 
points F each twice, and the points e each as many times as there are lines eF g through 
the point e, say each M times. (In the present case, where the curves e, F are identical, 
we have M= F — 2 or F— 3 according as the curve D is or is not distinct from the curve F ; 
in the cases afterwards referred to, the values may be F or F — 1 ; that is, we have always 
M=F, F— 1, F— 2, F— 3, as the case maybe.) We have to consider the several corre- 
spondences («, g), (a, F), (a, e); k is as before=Q ; and the form of the theorem is 
(g— 3C— x')+2(f— <p-<p')+ M (e — s — 0 = °> 
where the symbols denote as follows, viz. 
(a, g) have a (^, %f) correspondence and No. of united points = g, 
(a, F) „ (<p, <p') „ „ „ =f, 
(cz, e ) ,, (s , s ) ,, ,, ,, — 
so that the determination of g here depends upon that of f— <p — <p' and e — s — s'. 
6. The curve a might however have been identical with only one of the curves e, F ; 
viz. if a= F, but e is a distinct curve, then the equation will contain the term 
2(f— <p — <p'), but not the term M(e — s — s') ; and so if a=e, but F is a distinct curve, 
then the equation will not contain 2(f— <J5 — <p'), but will contain M(e — s — s') : it is to be 
noticed that in this last case we have M=F or M=F — 1, according as the curve D is 
not, or is, one and the same curve with F. The determination of (g) here depends upon 
that of f— <p — <p' or e — s — s', as the case may be. These subsidiary values f— <p — <p' and 
e — s — s' are obtained by means of a more simple application of the principle of corre- 
spondence, as will appear in the sequel*, but for the moment I do not pursue the 
question. 
Locus of a free angle {a). — Art. Nos. 7 to 14. 
7. I consider the case where a is a distinct curve zfce, #F, and where, as was seen, 
the equation is simply 
g-X-%'= °- 
* See post, Nos. 24 et seq. 
