SECT. II.] DETERMINATION &quot;OF COEFFICIENTS. 139 



Suppose the six following equations to be employed : 

 1 = a + b + c + d + e + f + &c., 

 = a + 3 2 Z&amp;gt; + 5 2 c +Td +9 2 e +H 2 /+&c., 

 = a + 3 4 & + 5 4 c + Td + 9 4 e + ll 4 / + &c., 

 = a + 3 6 6 + 5 6 c + Td + 9 6 e + ll 6 / -I- &c., 

 = a + 3 8 -f 5 8 c + 7 8 d + 9 8 e + ll 8 / + &c , 

 = a + 3 10 6 + 5 10 c + 7 w d + 9 10 e + ll 10 / + &c. 



The five equations which do not contain /are : 

 Il 2 =a(ll 2 -l 2 )+ Z &amp;gt; (H 2 -3 2 )+ c(H 2 -5 2 )+ J(ll 2 -7 2 )+ e(H 2 -9 2 ) ; 



0=a(ll 2 -l 2 )+3 6 6(ir-3 2 )+5 6 c(ll 2 -5 2 )+7 6 cZ(ll 2 -7 2 )+9 6 e(ll 2 -9 2 ), 

 0=a(ll 2 -r)+3 8 6(ir-3 2 )+5 8 c(ll 2 -5 2 )+7 8 ^(ir-7 2 )+9^(ll 2 -9 2 ). 



Continuing the elimination we shall obtain the final equation 

 in a, which is : 



a (ll 2 - 1 2 ) (9 2 - 1 2 ) (7 2 - 1 2 ) (5 2 - 1 2 ) (3 2 - I 2 ) = ll 2 . 9 2 . 7 2 . 5 2 . 3 2 . 1 2 . 



173. If we had employed a number of equations greater 

 by unity, we should have found, to determine a, an equation 

 analogous to the preceding, having in the first member one 

 factor more, namely, 13 2 I 2 , and in the second member 13 2 

 for the new factor. The law to which these different values of 

 a are subject is evident, and it follows that the value of a which 

 corresponds to an infinite number of equations is expressed thus : 

 32 52 7 2 92 , 



/Vrp 







_ 3 . 3 5.57.7 9.9 11 .11 

 ~ 2T4 476 6T8 8710 10TT2 



Now the last expression is known and, in accordance with 

 &quot;Wallis* Theorem, w r e conclude that a . It is required then 

 only to ascertain the values of the other unknowns. 



174. The five equations which remain after the elimination 

 of / may be compared with the five simpler equations which 

 would have been employed if there had been only five unknowns. 



