140 THEORY OF HEAT. [CHAP. III. 



The last equations differ from the equations of Art. 172, in 

 that in them e, d, c, b, a are found to be multiplied respec 

 tively by the factors 



n 2 -9 2 n -jT 2 ir- 5 2 ir-3 2 ir-r 

 &quot; iv * ~iY~ n 1 ~~Tr~ ir 



It follows from this that if we had solved the five linear 

 equations which must have been employed in the case of five 

 unknowns, and had calculated the value of each unknown, it 

 would have been easy to derive from them the value of the 

 unknowns of the same name corresponding to the case in which 

 six equations should have been employed. It would suffice to 

 multiply the values of e, d, c, b, a, found in the first case, by the 

 known factors. It will be easy in general to pass from the value 

 of one of these quantities, taken on the supposition of a certain 

 number of equations and unknowns, to the value of the same 

 quantity, taken in the case in which there should have been 

 one unknown and one equation more. For example, if the value 

 of e, found on the hypothesis of five equations and five unknowns, 

 is represented by E, that of the same quantity, taken in the case 



II 2 



of one unknown more, will be E- 2 . The same value, 



j. JL y 



taken in the case of seven unknowns, will be, for the same reason, 



11* -9* 13 -9&quot; 



and in the case of eight unknowns it will be 

 II 2 13 2 15 2 



E 



11* 9* 13* -9* &quot;15* -9&quot; 



and so on. In the same manner it will suffice to know the 

 value of b, corresponding to the case of two unknowns, to derive 

 from it that of the same letter which corresponds to the cases 

 of three, four, five unknowns, &c. We shall only have to multiply 

 this first value of b by 



5 2 7 2 9 2 



.. &c. 



5 2 -3 2 *7 2 -3 2 9 a -3 2 



