512] 
IN RELATION TO TWO TETRAHEDRA. 
205 
and consequently 
or, what is the same thing, 
1 .1.1 TJ 
x' ' y z flAx 
TJ 
' flAy 
. U 
flAz 
+ cot A — cot A' 
+ cot B — cot B' 
+ cot C — cot G', 
x : y : z = 
U+ A (cot A — cot A') fix 
V 
U + A (cot B — cot B') fly 
z 
U+ A (cot C — cot C') flz ’ 
where observe that the equations 
U+ A (cot A — cot A') fix = 0, 
U+ A (cot B — cot B') fly = 0, 
U + A (cot C — cot C') flz = 0, 
represent circles (B, C), (C, A), (A, B) containing the angles A', B', C'; and since 
A' + B'+C' — tt, these meet in a point 0. We may for convenience write 
, , BG CA AB 
x-.y-.z~ BCQ -. CA0 : ABQ , 
where BG = 0 denotes («=0) the line BO; BCO = 0 the circle through B, C, 0. And of 
course, in like manner, 
B'G' G’A' A'B' 
x : y : z- b , c ,q, ■ C ' A 'Q' : j/B'O'’ 
so that the points P, P' have a rational, or (1, 1), correspondence. 
Writing 
x' : y' : z = BG. CA 0. ABO : CA.ABO.BGO : AB.BCO.CAO 
X -. Y : Z 
suppose, X, Y, Z are quintic functions of x, y, z, and the curve in the first figure 
corresponding to the line ax' + fiy + yz' = 0 of the second figure is 
aX + /3Y + yZ= 0; 
viz. this is a quintic curve having dps. at each of the points A, B, G, 0, I, J. In 
fact, if for BGO we write BGOIJ, and so for the other two circles respectively, we have 
in an algorithm which will be at once understood X = BG. GAOIJ. ABOIJ, = (ABCUIJ)-, 
and similarly Y=Z, =(ABC0IJ)\ or the curve is {ABG0IJf = 0.