Thanks @Carl Love.
Yes, I found out how to create a 3D array, but I am wondering now if that is really the best way. It might be good for TSS, but I might have to change what I do depending on the meaning of each parameter. And I think that the notation is probably adding to my confusion. I am an engineer not a mathematician. The dots in place of the indices are shorthand for a prior summation in that dimension. For instance, I should only need to do a summation in the i dimension ultimately for SS_P. And the final term in each equation is the average of all my measurements.
By the way, A stands for Appraiser, and P stands for Part. I have n meaurements of each part from each appraiser. I suspect that the struggles I am experiencing explain why people just buy Minitab or some other statistical software instead of coding the math themselves. However, I would like to figure out the math.