237 
667] 
ON THE BICIRCULAR QUARTIC. 
this would, by virtue of the relations between dco, do) 1 , dw 2 , dco 3 , become 
8 VÔ _ 8 1 Vn i S 2 Vn 2 _ 8s VTf 
e x 2 +y 2 61 x x 2 + yZ + e ' X 2 + y 2 e '-’ xi + yi 
an equation not deducible from the relations which connect w, w 1 , w 2 , co 3 , and which 
therefore cannot be satisfied by the variable quadrilateral. 
26. The differentials of the formulas are, it will be observed, of the form Pda> 
8 do) 
(x 2 + y 2 ) V® ’ 
where V®, = ff + 6 . g + 6, is a mere constant, 
and 
viz. the form is 
COS ft) 
Sin ft) 
x ’ y Vÿqrg’ v^ïn9' 
S' 2 = {(/+ 6) x - a} 2 + {{g + 0) y - ßY - r î 
Vi (cos o) ff+ 6 — a) 2 4- (sin to fg + 0 — ß) 2 — y 2 
V® 
¿k /cos 2 to sm 2 to 
+ 
J + 0 g + 0 
do), 
which is, in fact, the same as Casey’s form in <£, equation (300), his cf> being 
= 90° - to. 
Writing as before v in place of his 0, the differential expression becomes simply 
= 8 dv: but 8 2 expressed as a function of v is an irrational function M + N V U, 
and 8 would be the root of such a function; so that, if the form originally obtained 
had been this form 8dv, it would have been necessary to transform it into the first- 
g 1 
mentioned form , in which 8 is expressed as a function of (x, y), that 
(^ + y 2 )V® r 
is, of ft). 
27. The system of course is 
dS = e8dv + e 1 8 1 dv 1 -I- e 2 8 2 dv 2 + e 3 8 3 dv 3 , 
dS x = e8dv — e 1 8 1 dv 1 + e 2 8 2 dv 2 + e 3 8 3 dv 3 , 
dS 2 = e8dv — e 1 8 1 dv 1 — e 2 8 2 dv 2 + e 3 8 3 dv 3 , 
dS 3 = e8du — e 1 8 1 dv 1 — e 2 8 2 dv 2 — e 3 8 3 dv 3 , 
where dv = -—-==., &c.; and this is the most convenient way of writing it. 
(,x 2 + y 2 ) v ® 
Reference to Figure. Art. No. 28. 
28. I constructed a bicircular quartic consisting of an exterior and interior oval 
with the following numerical data: (/ + 0 3 = 48, f + 0 1 = 56, /+ 0 O = 60, / + 0 2 = 80; 
g + 0 3 = -6, g + 0i = 2, g+0 o =6, g + 0 2 = 26),—not very convenient ones, inasmuch as