THE m AN D-CIRCUMSCEIBED TRIANGLE. 
399 
pass through a node or a cusp of this curve, viz. it is either a common tangent of the 
curves B=D = F and a=c=e (as in the figure, except that for greater distinctness the 
points c and e are there drawn nearly instead of actually coincident), or it may be a 
tangent to the curve B = D = F from a node or a cusp of the curve a=c=e\ we have 
thus the numbers 
Common tangent XY(Y— 1)(Y — 2), 
Tangent from node 2c5Y(Y— 1)(Y — 2), 
Tangent from cusp 2zY(Y— 1)(Y — 2) ; 
but (as we are counting intersections with the curve a~c—e ) the second of these, as 
being at a node of this curve, is to be taken 2 times ; and the third, as being at a cusp, 
3 times ; and the three together are thus 
(X+4S+6* )Y(Y— 1)(Y— 2), 
= {2#(#—l)— X}Y(Y— 1)(Y— 2). 
The reductions 1°, 2°, 3° altogether are 
2x(x- 1)Y(Y— 1)(Y— 2) 
— 2x{x— l)y{Y—2) 
+ 2x(x — 1 MY — 1 ) 
+ 24r— 1)Y(Y— 1)(Y— 2) 
— XY(Y — 1)(Y— 2), 
which is 
= 4lx(x— 1 )Y(Y — 1 )( Y — 2) 
+2x{x-l)y 
— XY (Y — 1)( Y — 2) ; 
and subtracting from the before-mentioned number 
2x 2 (x — 1 ) Y( Y — 1 )( Y — 2 ) 
+x-(x-l )y, 
the required number of positions of the angle a is 
= 2x(x- l)(x- 2) Y(Y - 1)(Y - 2) 
+ y#(# — 1 ) (# — 2 )«/ + X Y( Y — 1 )( Y — 2 ) . 
The number of triangles is on account of the symmetry equal to one-sixth of this 
number. 
