414] 
ON POLYZOMAL CURVES. 
511 
97. Let (a,, b 1} c x , d,) be the corresponding quantities to (a, b, c, d), viz., 
a, : b, : c, : d, = Bfi^ 1 : — C 1 D 1 A l : D 1 A 1 B 1 : — A^B ± G^; we have 
3^ : bj ! C] ! d] — 
giving rise to a similar set of forms 
/3, 7'» 1 
: - 
7. /3', 1 
S, a', 1 
: - 
a, S', 1 
7 > /3', 1 
8, a', 1 
a, S', 1 
/3, 7, 1 
S, a', 1 
S', 1 
/3, y, 1 
7, 0', 1 
a, — 
, — ac' + ha', a g + h'a, 
b, = 
-c'b-g'h, . , —f'b — g'f, 
Ci = 
b'c + h'g, -f'c + h'f, 
di = 
g'c + h'b, — li'a + a'c, — a'b — g'a, 
and leading to 
lL„ = _y. K = a l o ft 
a'c' 1 c't/ 1 a'/' 1 a'g' 
so that the equation 
0 , m i , n i , 
— 4* T~ + — 4* j" 
Sij Dj Clj 
is transformable into 
ac eg aj ag x 
f'9+9'f, 
= 0, 
98. Let A, By G, D, be, as above, points on a circle; (A 1} D,) and (B 1} Gj) the 
antipoints of (A, D), (B, G) respectively. Write 
21 = (£ — az ) (77 — a'z ), 21, = (£ — az ) (77 — S'z ), 
93 = (£ - /3s) (v - /3’*), 93, = (f - /3z) ( v - f/s), 
g = (£ - l z ) (v - l z \ ®1 = (I - ys) (v - (3'z), 
Q = (% - &z)(r} - S'z), Di = (£ -$z)(ri - a'z ) ; 
then we have identically 
(S - a) (S' - a') 23 = (/3 - S) (/3' - S') 81 + (/8 - a) (/S' - a') 5) -08-S)(/8'-a08k-G8-a)08'-S')$ 1 , 
(8 _ a ) (S' _ o') g = ( 7 - S) (7' - S') 21 + (7 - a) (7 - «') 5> - (7 ■-S)( 7 ' - o')8k-(7 -«KV -S') 5>i, 
( 8 _ a) (8' - a') 23, = (/8 - S) (7' - S') 21 + (/3 - a) (7' - a') 2) -(/3-S)( 7 ' -a') 21,-03-a) (7' -S')D„ 
(8 _ a ) (8' - a') g, = (7 - S) (/3' - S') 21 + (7 - «) (/S' - a') D -(7 -8)( i S'-a')2l 1 -( 7 -a)(/3'-£')£>„ 
or, in the foregoing notation, 
ff'%3 = gg'% + cc'2) + ^c'21, + c^'iDu 
//g = hh'A + 66'2) - M'21, - 
y7'23, = <//i'2l - c&'$ - ^'21, + ch'fy, 
//'g, = %'2l - 6c'D + /¿c'2lj - %'£),.