454 
PROFESSOR CAYLEY ON THE BICIRCULAR QUARTIC. 
24. But we have B'— B,= — &c. : and hence, putting for shortness 
x 2 + y 2 r & 
(x*+f) a/©’ (zf + yf) V©! 5 (* s a + y 2 *) V@ 2 ’ (^#W 3 ==P ’ P ” ^ Ps ’ 
dS -j-<2$ 3 =+2g P dco, 
dSt-dS =-2 gl PA, 
<2S 2 — <?Sj = — 2g 2 P 2 <?ft/ 2 , 
^S 3 — <ZS = — 2g 3 P 3 6?<y 3 , 
and consequently 
<7S = gP dco -f- g z P 1 dco l -j~ g 2 P 2 dco 2 + s 3 P 3 ^ 3 , 
£ZSj=gP(7y — g 1 P 1 c7iw 1 “h g 2 P 3 dco 2 ™f~ £ 3 P $dco 3 , 
• dS 2 =ePdoo — gjPjQfft^ — g 2 P 2 (fot> 2 -f- g 3 P 3 <7<y 3 , 
(ZS 3 = gP^cy — gjP^^j — g 2 P 2 (7a> 2 — g 3 P 3 c7y 3 , 
which are the required formulae for the elements of arc. 
25. The determination of the signs has been made by means of the particular figure ; 
but it is easy to see that the pairs of terms could not for instance be dS — dS 3 , dS t —dS, 
dS 2 — <7S„ dS 3 — dS, or any other pairs such that it would be possible to eliminate 
dS, dJS„ dS 2 , dS 3 , and thus obtain an equation such as 
sPdco — J— g 1 P 1 6 Z &/ 1 -J-g 2 P 2 (7iy 2 -J-£ 3 P 3 £Ziy 3 =0 ; 
this would, by virtue of the relations between dco, doo y , dco 2 , dco 3 , become 
sya_ s, vn x , s 2 ya 2 s 3 Vn 8 _ 0 
1 + V \'' "xi+y* 3 ^+Vz ’ 
an equation not deducible from the relations which connect co, co n co 2 , co 3 , and which 
therefore cannot be satisfied by the variable quadrilateral. 
26. The differentials of the formulae are, it will be observed, of the form P dco 
i dco 
(x 2 + y 2 ) \/ 0 
/75 . cosai smw 
where v 0, = s / j- . g-\-Q is a mere constant, x, an d 
& 2 =((/+^- a ) 2 +((^+%— 0 ) -r 2 
viz. the form is 
\/ (cos co \Zf+Q — «) 2 + (sin co *dg-\-b— (3) 2 — y 2 
, — /cos 2 co „ sin 2 co \ 
^ e -( 7iT+7f»-) 
dco, 
which is, in fact, the same as Casey’s form in <p (equation (300), his <p being =90°— co). 
Writing as before v in place of his 6, the differential expression becomes simply =ldv. 
