CHAP. XX.] PROBLEMS ON CAUSES. 339 



For instance, if n = 2, we shall have 



x l9 x Z9 &c. standing for 1 - x, 1 - x 2 , &c. ; whence 



Z 



+ constituents whose coefficients are -. 



This result agrees, difference of notation being allowed for, with 

 the developed form of z in Problem I. of this chapter, as it evi- 

 dently ought to do. 



10. To avoid complexity, I purpose to deduce from the above 

 equation (6) the necessary conditions for the determination of 

 Prob. z for the particular case in which n = 2, in such a form as 

 may enable us, by pursuing in- thought the same line of investi- 

 gation, to assign the corresponding conditions for the more gene- 

 ral case in which n possesses any integral value whatever. 



Supposing then n = 2, we have 



V= 



n , 



Prob. 



the conditions for the determination of x l9 t l1 &c., being 



Xi X z ^ tz + X^ ~X Z ti~fz + Xi Xz 7i? 8 + *1 ^2 7] TZ 



Ci 



\ MZ t\ t z + ^i X-iti tz + Xi Xz trfz 



Xi Xz ti tz + Xi Xz 



Divide the members of this system of equations by ^ x x 2 7i7 2 , 

 and the numerator and denominator of Prob. z by the same quan- 

 tity, and in the results assume 



Xi t\ Xz tz Xi X z x^v 



^-=r = wii, =-= = ?w 2 , = !, = w 2 ; (7) 



Xi t\ Xz tz Xi Xz 



z 2 



