a System of Equations . 3 7 1 



in which the several sets of/, m 9 n; X, /z, v can be expressed 

 without difficulty in terms of the several values of Vf, Vg, Vh. 

 Let the above equations be written under the form 



PF=o 

 QQ'=o 

 RR'=0. 



Since the given equations are perfectly general, it is readily 

 seen that the equations 



(P = P = 0) (Q = Q' = 0) (R = R' = 0) 



will severally represent pairs of opposite sides of a quadrangle 

 expressed by general coordinates x, y 9 z; so that one of the 

 two functions R, R' will be a linear function of P and Q and 

 also of F and Q', and the other will be a linear function of P 

 and Q' and also of P' and Q*. 



In order to solve the equations, we need only consider two 

 such pairs as PP' = QQ' = 0; we then make 



P = Q=0, 

 or 



P=0 Q' = 0, 

 or 



F=0 Q = 0, 

 or 



F=0 Q' = 0. 



Any one of these four systems will give the ratios of x : y : z; 

 and then, by substitution in any one of the given equations, 

 we obtain the values of x, y, % by the solution of an ordinary 

 equation of the nth degree. The number of systems x, y 9 z 

 is therefore always 4w. 



The equations connected with the solution of Malfatti's 

 celebrated problem, " In a given triangle to inscribe three 

 circles such that each circle touches the remaining two circles 

 and also two sides of the triangle," given by Mr. Cayley in 

 the November Number for 1849 of the Cambridge and Dublin 

 Mathematical Journal, to wit, 



by 2 + cz 2 + 2fyz=d 2 . aipc-f*) = A 

 cz 2 + ax 2 + 2gzw = 6 2 . b{ca —g 2 ) = B 

 ax 2 + by 2 + 2hxy = <9 2 . c(ab - h 2 ) = C, 



* "Were it not for this being the case, the number of solutions would be 

 n times the number of ways of obtaining duads out of three sets of two 

 things, excluding the duads forming the sets, i. e. the number of solutions 

 would be \°Zn in place of 4w, the true number. 



2 B 2 



