If either or both of the sum and sum of squares of n variates is(are) minimal sufficient, inferential procedures are based on their joint density. In cases where the variates are non-negative the derivation of this joint density is non-trivial, and no closed-form expression for it seems to be known. Using a differential-geometric approach, we derive this joint density for the class of exponential models in which either or both of these statistics is(are) minimal sufficient. The results have numerous applications; one of these, the censored normal model, is considered briefly.