510 
ON POLYZOMAL CURVES. 
[414 
and we may write 
viz., the 
portional 
a — * , 
ah' — a'h, 
ag' - a'g, gh! - g'h, 
b = bh' - b’h, 
• 
bf'-bf, lif — hf, 
0 
II 
r 
0 
cf-cf, 
> fg' -fg> 
fi 
ll 
0 
CH 
1 
ch 
ad — a'c, 
ba' — b'a, . , 
expressions in the same 
horizontal 
line are equal, and a, b, 
to the expressions in the four lines respectively. 
c, 
d are pro- 
95. I say that we have 
viz., this will be the case if 
bh af ah ’ 
bc'& = kg' d, 
ad b = hf d, 
=fg d, 
and selecting the convenient expressions for a, b, c, d, these equations become 
bd {gli' — g'h) = g'li (cb' — db), 
ad {hf — hf) = f'h (ad — a'c), 
ct'b(fg'-f'g)=fg'(ba'-b'a), 
viz., these equations are respectively bgc'li = b'g'ch, cha'f' = dh'af afb'g' = a'f'bg, and are 
consequently satisfied. It thus appears that the equation 
is transformable into 
l m n 
abed 
P _ 
= 0 
oTi.cy 
ah bh 
af' 
af 
m + —^ n +‘ / —y- I) = 0, 
_/y 
ab 
which is of course one of a system of similar forms. 
96. Take (A lt A) the antipoints of (A, D); (B 1} G x ) the antipoints of (B, C); 
or say that the circular coordinates of A u B u C x , B 1 are (a, S', 1), (/3, y, 1), (7, /3', 1), 
(8, a', 1) respectively; the points A x , B x , C u D x are, as above mentioned, on a circle, 
the condition that this may be so being in fact 
1, a, S', aS' 
1, ß, f, ßy 
1, 7, ß', yß' 
1, 8, a , Sa.' 
= 0, 
af : bg : ch — af' : b'g : dh'. 
equivalent to