Help with Uspensky "sum of three squares"

I need some help with Uspensky's proof on the sum of three squares found in "Elementary Number Theory" by Uspensky and Heaslet. I have gotten to the end of the proof and do not understand a few statements found at the end. I was wondering if anyone was familiar with the proof. In it he defines to be the total solutions to

With and all odd.

Let be the number of representations of as the sum of three squares.

He shows

Which I understand where that comes from. What I am confused on, is he claims that since this identity is true for all odd values of that we must have

Why must we have that? Any help would be greatly appreciated.