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386 Rh CHAPTER XLII.

UNDETERMINED COEFFICIENTS.

FUNCTIONS OF FINITE DIMENSIONS.

481. In Art. 105 it was proved that if any rational integral function of 2 vanishes when x=a, it is divisible by x-a.

Let

be a rational integral function of x of x dimensions, which vanishes when x is equal to each of the unequal quantities

Denote the function by f(x); then since f(x) is divisible by x-a_1, we have

the quotient being of n-1 dimensions. Similarly, since f(x) is divisible by x-a_2, we have

the quotient being of n-2 dimensions; and

Proceeding in this way, we shall finally obtain after n divisions

482. If a rational integral function of m dimensions vanishes for more than n values of the variable, the coefficient of each power of the variable must be zero.

Let the function be denoted by f(x), where