E Q U 



be the equations ; multiply the first by a, the second by a', and so on ; 

 then by addition, 

 ma 4- m> a' 4- &c. -f- (as 4- a'2 4- &c.) JP + (a 6 -f ' &' 4- &c.) y 4- 



(a c 4- a' c' 4- &c.) z = o, 

 Similarly 

 (w 6 4- ' 6' 4- &c.) -I- (ab + a' b' + &c.) x -f (ft, + i'a + &c.) y -f- 



(b c 4- b' c' 4- &c.) x 4- &c. = o, 

 (w c + ;' c x -f- &c.) + (ac + a' c' + &c.) ^ + (6 c + b' c' -f- &c.) ^ + 



(c 8 + c'* H- &c.) 5r + &c. o, 

 &c. 4- &c .................................................................................. o. 



By this means as many equations are formed as there are unknown 

 quantities, and from them #, y } z, &c. may be determined. 



The method applied to the example in the preceding" article gives the 

 reduced equations 



884-27^4-6^ = 0, 



70 4- 6 x 4- 15 z o, 



107 4- y 4- 51 x = o. 

 From whence x 2.470, y 3,551, z = 1.916. 



The above mode of reducing the linear equations, which is called the 

 Me f hod of Least Squares, Avas invented by Gauss. 



EQUATION of Payments. 

 Common rule. 



Let p and p' be the sums due at the end of the times n and n' * = 

 equated time 



p n -\- p> n' 

 then x = - -, - . 

 p-rp' 



i.e. equated time is found by multiplying each sum by the time at which 

 it is due, and dividing by the sum of the payments. 



This rule is erroneous in principle, being founded upon the supposi- 

 tion that the receiver gains interest upon the latter sum by receiving it 

 before it is due ; Avhereas in fact he ought only to gain the discount. In 

 most questions, however, that occur in business, the error is so trifling, 

 that the above rule Avill always be made use of as the most eligible me- 

 thod. 



Correct rule. 



Let r interest of 1. for one year, the rest as before, put 



= ^ prnn> + p>n> + pn = 



p r p 



a V 4 b 





105 



