CHAP. VIII.] OF REDUCTION. 121 



lent to the collective interpretations of the several equations from 

 which it is derived. 



PROPOSITION III. 



5. If V l = 0, F 2 = 0, fyc. represent any system of equations, the 

 terms of which have by transposition been brought to the first side, 

 then the combined interpretation of the system will be involved in the 

 single equation, 



V* + F 2 2 + $c. = 0, 



formed by adding together the squares of the given equations. 



For let any equation of the system, as Fj. = 0, produce on de- 

 velopment an equation 



a\t\ + a z t 2 + &G. = 0, 



in which t l9 t z , &c. are constituents, and 15 a Z9 &c. then* corres- 

 ponding coefficients. Then the equation Fj 2 = will produce 

 on development an equation 



<Zi 2 i + 2 2 *2 + <& = 0, 



as may be proved either from the law of the development or by 

 squaring the function a l t 1 + # 2 2 , &c. in subjection to the con- 

 ditions 



*1 2 = *1, * 2 2 =*2, *1*2 = 0, 



assigned in Prop. 3, Chap. v. Hence the constituents which 

 appear in the expansion of the equation F x 2 = 0, are the same 

 with those which appear in the expansion of the equation V l = 0, 

 and they have positive coefficients. And the same remark ap- 

 plies to the equations F 3 = 0, &c. Whence, by the last Propo- 

 sition, the equation 



F, 2 + F 2 2 + &c. = 



will be equivalent in interpretation to the system of equations 

 F! = 0, F 2 = 0, &c. 



Corollary. Any equation, F= 0, of which the first member 

 already satisfies the condition 



F 2 = F, or F(l - F) = 0, 



