170 THEORY OF HEAT. [CHAP. III. 



and we omit from each equation all the terms of the second 

 member which follow the first m terms which we retain. The 

 whole number m being given, the coefficients a, b, c, d, e, &c. have 

 fixed values which may be found by elimination. Different 

 values would be obtained for the same quantities, if the number 

 of the equations and that of the unknowns were greater by one 

 unit. Thus the value of the coefficients varies as we increase 

 the number of the coefficients and of the equations which ought 

 to determine them. It is required to find what the limits are 

 towards which the values of the unknowns converge continually 

 as the number of equations increases. These limits are the true 

 values of the unknowns which satisfy the preceding equations 

 when their number is infinite. 



208. We consider then in succession the cases in which we 

 should have to determine one unknown by one equation, two 

 unknowns by two equations, three unknowns by three equations, 

 and so on to infinity. 



Suppose that we denote as follows different systems of equa 

 tions analogous to those from which the values of the coefficients 

 must be derived : 



a^ = A^ a a + 26 2 = A a , a 3 + 2& 3 + 3c 3 = A z , 



3c 4 



3c 5 



&c. &c ......... . ................ (b). 



