452 
PROFESSOR CATLET ON THE BICIRCULAR QUARTIC. 
we find 
{(?+« XP-M +(/+ 0 («-«> +(t-b,)((f + 6 'm,+<g+'j )«,)} 
+^\(s+W-l3,)y,+(f+^-^M6-mf+^+(9+^yh)\=Oi 
viz., dividing by Q — this becomes 
£?CO 
~' /s Vii~' /Q vs; =0 ’ that is 
or, completing the system, we have 
dco — dco j 
V© Vo V©i Vo, 
:0: 
date, 
■dco o 
V© Vo V©i Vo, V© 2 Vo 2 V© 3 Vo 3 ’ 
which are the differential relations between the parameters «/, a/„ a 2 , <a 3 , or (a?, ^), (a:,, 
(*«&)> fey 3 )> 
22. From the equations X=a-f-B/a?, Y=/3+R , y, we found 
dX= 
R ’dco 
Vo V© 
{Y -(g+6M, 
dY= 
R 'dco 
Vo V© 
\X-(f+Q)x\; 
the new values, X=a,-|-Il 1 # 1 , Y = /3, -J- Itiyi give in like manner 
(ZX =-Jfwj Y -^+^S’ 
and in virtue of the relation just found between da and dco x these two sets of values will 
agree together if only 
K'{Y— fo+%}=B,{Y- (g+O^h 
E'{X-(/+S>}=K,{X-(/+« 1 > 1 (. 
These are easily verified : the first is 
^Y-(^+S)(Y-/3)=(E'-5+^)Y-(^+5 1 (Y-/3 1 ), 
viz. this is (g + Q)p— (g+Q 1 )^ 1 =0, which is right; and similarly the second equation 
gives (/+$)a — (yH-$i)«i = 0, which is right. 
From the first values of dX, dY we have, as above, 
s'R'8 dco 
db= Vo V©’ 
