After some remarks about the analogy between the classical gamma-function and Gaussian sums over finite fields a complete, very short explicit proof is given of an identity expressing a certain sum of products of Gaussian sums as a product of Gaussian sums. This identity is an analogue of the classical Barnes' first lemma for the gamma-function. Four multiplicative characters of a finite field are concerned; the usually necessary restrictions on the triviality of certain products of these characters are avoided by the use of corrective terms. References are given for other approaches of this identity.\par In  a parallel proof is given for the classical identity and its finite analogue; the status of this reference has meanwhile changed from ``preprint'' to ``published'': Can. Math. Bull. 36, No. 3, 273-282 (1993; Zbl 0803.33001), the status of reference  has changed from ``to appear'' into ``published'': Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 179-188 !