514 
ON POLYZOMAL CURVES. 
[414 
and for each of which we have 
L m, n, «1 7 , 
-+i—I hV = 0, i p — ip m n — mn. 
aj Cj- r 
103. Consider, in particular, the case where p — 0; the relation 
l m n p 
-+t-4--4-^ = °, 
abed 
, aq' a'h 
l =-bf m ~Tf n - 
(<cc'm 4- bb'n) 6 4- cb'n6- 4- bem — 0, 
(c6 4- c'm) (b'n6 4- b) — 0, 
here becomes 
The equation in 6 is 
viz., this is 
giving 
or else 
Since in the present case l 1 p 1 = 0, we have either = 0, or else p x = 0, and as might 
be anticipated, the two values of 6 correspond to these two cases respectively, viz., 
proceeding to find the values of l 1} p 1 , the completed systems are 
0 = 
b 
bn 
cm 
c 5 
m 1 = -- , 
Ul ~~T 
<9 = 
c'm 
c’m 
b'n 
~ b'n ’ 
m, = - , 
$ 
II 
1 
o^| 
0 
b j a ( , \ bn cm 
= -- , k = (ccm-bbnj, m 1 = - — , n x - - ^ , p x = 0, 
6 ~ b'n 1 1 ° 
so that for the first system we have 
bn 
cm 
V ’ 
w >=- b 
c'm 
v b'n 
~b r ’ 
n ' = ~Y 
b'cf 1 
- + ^ + - = o, mp h = mn, - № + m33 4- n<£ = - Wi + + nfa, 
a 2 bj c, 
and for the second system 
4- — 4- = 0, m\n\ = mn, — Z214- m33 4- n(S = + m\33 X 4- w\(5i. 
D x C x Cl! 
104. The whole of the foregoing investigation would have assumed a more simple 
form if the circular coordinates had been taken with reference to the centre of the 
circle ABGD as origin, and the radius of this circle been put = 1; we should then 
have cl = ^, &c., and consequently 
a' = — -5- cl, b' = — — b, c' = — “5 c, = 9' = — -¡5*9 > bf = —\h] 
fiy ya. a/3 ’ J a8 J J /38^’ y8 
but the symmetrical relation of the circles A BCD and A 1 B 1 C 1 D l would not have been 
so clearly shown.