PEOEESSOE CAYLEY ON THE BICIECULAE QITAETIC. 
453 
and the second values give in like manner 
dS= 
gjEjSj dco j 
where g, is= ±1. It will be observed that we have in effect, by means of the relation 
(/3 — /3,)(#— #,) — (a — «i)(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'C'D', then dS is the element of arc AA' ; and writing in like manner 
dS 2 , dS 3 for the elements of arc BB', CC', DD', we have, of course, a like pair of 
values for each of the elements ^S 1; dS 2 , dS 3 . 
Formulae for the elements of Arc dS, dS„ d S 2 , dS 3 . — Art. Nos. 23 to 27. 
23. The formulae are 
tfS 2 =g'R'& 2 - 
dX—s'MX 
dco { 
Vlii V©! 
dlOy 
_js«a - 
V0 3 V© c 
dco j 
VEt, V©i’ 
dco 2 
™ vo 2 v© 2 ’ 
dco 3 
=sj&$i 
=£ 2 R 2 eS 2 
= £ 3 R 
=g R 
3 3 3 Vn 3 V©3 
dco 
\/n V© * 
where the g’s each denote +1. Supposing as above that y 2 is negative, but that y 2 , y 2 , y* 
are positive; then R', R have opposite signs: but R', R, have the same sign, as have 
also R 2 and R 2 , and R 3 and R 3 . We may take &, h 2 , and as each of them positive : 
,i r dco dco dcoc, dco 3 
the 8I ? ns of W7S’ 75^®;’ VSs V* 
hence to make dS, dS t , dS 2 , dS 3 all positive, 
£ , £j , £ 2 , £ 3 , Si , S 2 , £ 3 , 2 , 
must have signs of 
or else the reverse signs: hence in either case s'=— s, g' 1 = g 1 , g' 2 =g 2 , s' 3 = e 3 ; or the 
equations are 
dS = — gR'& 
dco 
s/£l V© 
dco , 
rSjR^l 
C?c0j 
*■= 
VOj V©, 
2 
VXi 2 V© 
dS 9 = g 3 R$ 
dS. 
dcoc, 
3 2 2 Vn 2 V© 
dco ? 
3 V ^3 
£ 3ll'3^3 ./(=r /fiT 
vfl 3 v©; 
= £ 3 R 3 ^ 3 
dco 0 
VO, v©3 
dco 
\/n v©' 
