Theorem on Fourth Powers 319 Extensions of Waring ' S T H E O R E M on Fourth Powers

is said to be of order n and weight &1+&2+ • • • + a n . Since ax* = x*+ • • • +^ 4 , to ^ terms, a form of weight w is equal to a sum of w biquadrates. But 79 is not a sum of fewer than nineteen biquadrates. Hence 19 is the minimum weight of a form (1) which represents all positive integers. Let ƒ be a form (1) which represents p, and let ai = r+s. The form g = (r, s, a2, • • • , an) shall be said to be derived from ƒ by the partition of ai into r+s. If we give to the first two variables in g the same value Xi as was employed in ƒ — Pi we see that also g represents p. Hence any form derived from ƒ by partition represents every integer which can be represented by ƒ (and usually represents further integers). Write a = 2, & = 3, c = 4, • • • . If a positive integer m can be expressed as a linear combination of 1, a, 6, • • • , with integral coefficients ^ 0 whose sum is ^ 1 9 , in one and only one way, m shall be called a simple number. In case there are exactly two such expressions, m shall be called a double number. Similarly for a triple or k-fold number.