ON THE BICIRCULAR QUARTIC. 
235 
667] 
in virtue of the relation just found between dco and dco lt these two sets of values will 
agree together if only 
R ' i Y - (9 + e ) y) = R i i Y ~ (9 + 00 2/i}> 
R' {X - (/+ 6) = R, {X - (/+ 9,) 
These are easily verified: the first is 
R'Y -(g + 6)(7-/3) = (R - 6 + 0\) 7- (<7 + ^(7- &), 
viz. this is 4- 0) /3 — (<7 4- ^1) & = 0, which is right; and similarly the second equation 
gives (/+ 9) a —(f+ 0j) oil = 0, which is right. 
From the first values of dX, dY, we have, as above, 
_ e'R'8 dco 
Vo V® 5 
and the second values give in like manner 
€iR>i&i do)i 
Vi^V®, ’ 
where e x is = + 1. It will be observed that we have in effect, by means of the relation 
(/3 — (x - x-i) — (a — a x ) (y — y x ) + (9 — 9 t ) (xy x — x-y) = 0, proved the identity of the two 
values of dS. 
Considering the quadrilateral ABCD, and giving it an infinitesimal variation, so as to 
change it into A'B'G'D' t then dS is the element of arc AA'; and writing in like manner 
dS lt dS 2 , dS 3 for the elements of arc BB', CG', DD', we have, of course, a like pair of 
values for each of the elements dS 1} dS 2 , dS 3 . 
Formulce for the elements of Arc dS, dS 1 , dS 2 , dS 3 . Art. Nos. 23 to 27. 
23. The formulae are 
dS = e'R'8 
dS 2 = e 2 'R'8 2 
dco 3 ^ g dco 
dS 3 = e 3 'R 3 '8 3 
Vo 3 V® 3 Vo V® ’ 
where the e’s each denote + 1. Supposing as above that 7 2 is negative, but that 
7i 2 > Y2 3 > 73 2 are positive; then R', R have opposite signs: but Rf R x have the same sign, 
30—2