266 
MR. A. CAYLEY’S ANALYTICAL RESEARCHES CONNECTED lYlTH 
or finally, if 
+ y" - 45^ (y = / 3^ + 7" - 45(7 + 
which is in fact the case. 
Writing the equations for 
in the form ^+^= 5 ^^ 
•^“S=^:r;p(o'-2/3s), 
and substituting in U=^|(^A— j)(a|+/3;(+yQ — ^A+i)s*(|+Q|, 
we have U=^{(7-2|3i)(<i5+/3!i+yQ— 5=(5+f)} 
H" 2-yy + 207)1 + (y — 2^/3) +(— / 3 + 2^7 4- 45(J/3) ^} . 
And consequently, multiplying by 
Is=2VW+WWTWWW^ 
we have A'^ + - C^^/A"^ + B"^ - C"^ 
= §\/ { ( “ ^ 4- 2^y + 2ip7) 1+ (7 — 2.9^);? + ( — /3 + 2^7 + 45®/3) ^ } , 
or collecting the different terms 
(«V'^-(3'(3")l+(/S''+y''/3'+2f/3'^")^ + (/3'/3''+yy')?+^'(<t?^-/3>,+rO 
— §\/ (y+2<pf3')(y"+2?)f3")|3'f3"'){( — /3+2sy+2^7)H-{7 — 2s/3);j-i-( — /3+2i’y+t4‘£^)Q 
which, combined with the first equation written under the form 
(»|+/3,+yO’-S“[(5-OHfl=0, 
determines the ratios of |, ??, i. e. the values of Y+Z and YZ. 
The system of equations 
(/+2«yg)+x/a(Y+z)+/Yz-yfcyr+V\/T+z-=o 
(^+2«x/B) +\/33(Z+X) +gZX-x/m^/T+Wyr-fW=o 
(A+2^yC)+^/C(X+Y)4■/^XY-y^^/T+X■^/T+Y'=0, 
where 
on which depends the solution of Steiner’s extension of Malfatti’s problem, is at 
once seen to belong to the class of equations treated of in the preceding section, and 
