414] 
ON POLYZOMAL CURVES. 
513 
where we see that the two equations are equivalent to each other and to the 
equation 
l x m, n x p x . 
- + vr + — +4 = o. 
a x b x Cj dj 
It thus appears that the quantities l x , m x , n x , p x , must satisfy this last equation. It 
is to be observed that the first and second equations being, as we have seen, equivalent 
to a single equation, either of the quantities m x , n x , may be assumed at pleasure, but 
the other is then determined; the third and fourth equations then give l x , p x ; and the 
101. Now writing 
ff'li ——9 (c'm + b'm x ) + h (b'n + cn x ), 
f/Pi = ~c (gm- h'm x ) + b (h'n-g'n x ), 
and 
ff'p = c (c'm + b'nii) + b (b'n + cn y ), 
ffl = 9 (g'm ~ h'rth) + h (h'n - g'n^, 
we find 
pp {lp x — lp)= — (bg + ch) [(c'm + b'm x ) (h'n — g'n x ) + (g’m — h'm x ) (b'n + c'wj)], 
= (bg + ch) (b'g' + c'h' ) (mgix — mn), 
= (Miff (m x n x — mn), 
that is 
ff (hpi — Ip) = (Mi (m x n x — mn) 
viz., this equation is satisfied identically by the values of l x , m x , n x , p x determined as 
above. 
102. Hence if m x n x — mn, we have also l x p x = Ip, and we can determine m x , n x> so 
that m x n x shall = mn, viz., in the first or second of the four equations (these two being 
equivalent to each other, as already mentioned), writing m x = 6n, and therefore n x = ^ m, 
we have 
—ffl + gg'm + lilt n — glind — g'hm ^ = 0, 
cc'm + bb'n — ff p + cb'nO + be'm - = 0, 
which are, in fact, the same quadric equation in 6, viz., we have 
—ffl + gg'm + hh'n _ gh' _ g'h 
cc'm + bb'n — ffp cb' be' 
The final result is that there are two sets of values of l x , nh, n x , p x , each satisfying 
the identity 
- № + + n& - pQ + l$L x — — W!®! + p x 9ö x = 0, 
C. VI. 
65