178 ME. ROBERT FINLAT ON 



. . PM..PM,...PM„- ^ j^^ _^ ^^.^ ^j ^ ^^,^^ _^^ _^ ^^,^^ 



^ ^' (9) 



This equation shows that the function u' has a fixed ratio 

 to the continued product of the perpendiculars PM^, PM.^, 



PM, PM». Denoting this ratio by M, equation (9) 



gives 



M'=A;(l + ?w;)(l + w;)(l + m;) (1 + w/) (10); 



or, by performing the multiplication 



M''^ A,* {1 + irn: + m^ + m: + &c.) 



+ {m^ m^ + m* m^ + m^ m^ + &c.) 



4- (»i,'w,'m3»+ &c.) + &c.} (11). 



The values of these symmetrical functions of the roots 

 »»,, m^, &c. may be very easily obtained by constructing the 

 equation whose roots are m^, m*, m^, &c. For the sake of 

 simplicity let us consider the case of » = 5 ; then equation 

 (7) becomes 



A^m'-\- A^m*-{- A,m' + A^w" + A,w-|- A„= o. 

 If we assume tn^ = z this becomes 

 ssi{A^z' + A,z + AJ + A^z' + A^z + A^ = o; 



hence, by clearing the radical, and arranging according to the 

 powers of z, we get 



AX+(2A,A^— A/)s*+ (A,»+ 2A,A^— 2A,AJs' 



+ (2A3A— 2A,A— A;);2*+(A;— ^A^AJis— a; = o. 



Now since w,', to/, mj*, &c. are the roots of this equation, 

 we shall have 



m: + m: + &c. = (A;— 2 A, A3) : A;\ 



m*m^'+m;'m^'+ &c. = (A;— 2 A, A, + ^ A. AJ : A,*, 



»»>/<+ &c.= (A; + SA.A,— 2A3AJ : a;, &C.&C. 



