In-and-circumscribed Triangle. 517 



Hence reducing the function on the right-hand side, and putting 

 *f + V? + »£+*? + y? + z*= 1, we have 



~ = a?/*a?i« + ja^e + c z n 4 2 6 



+ (£ 2 « 4 -2c 2 /» *n*) ! , 1 %4+ (c 2 / 4 - 2a W)r, V + (a*m 4 - 2b*l*m*)x*y l ' i 



+{;*(5-c) 2 +jw 4 (c-«) 2 +» 4 (fl-i) d 



+ 2mV(fe - ca - ab) + 2n*l*( -bc + ca-ab) 



+ 2l*m*(-bc-ca + fl6)}*,Vi V 



The value of © might probably be expressed in a more simple 



form by means of the equations /#, 4- my l + nz x = and 



/ 2 6e + »t 2 ea + ?i 2 «£ = J even without solving these equations; 



but this I shall not at present inquire into. 



Recapitulating, I, m, n are considered as functions of w v y lf z x 

 determined (to a common factories) by the equations 

 Ix j + my x + nz l = 

 Fbc+m*ca+n*abs=0. 

 © is determined as above, and then writing 



* = ax* + by* + cz*-a{x* + y* + z*) 

 P=ax* + by* + cz*-b{x* + y* + z*) 



7=ax* + by* + cz 1 *-c(x* + y* + z*), 

 we have 



■*' 2 = %i-i(/3rc 2 + 70 



y 2 = ©^,( 7 / 2 + a « 2 ) 



- 2 = ©^ 1 y 1 ( a m 2 + ^ 2 ). 

 And these values give 



lx 2 + my % +nz^=0 



ax* + by* + cz*=ax* + by* + cz*. 



In connexion with the subject I may add the following trans- 

 formation, viz. if 



Then reciprocally 



8 •#*=- s/3a(y'-z')+ ^W^-^)(^+y'"+^)+2u(y'z'+z'x'^yJ. 



A,8 ° ++.p++=*J*-4J*+0 



{3(x* + y* + z*- yz -zx-xy)=cc (ar* + y* + a* - ,/„-' - .-'a' - x'y') . 



