Computer generated pseudo random numbers were used to simulate drawing 1000 pairs of samples ofN1,N2= 5, 10, 20 from bivariate populations normal (O, σi2I) having σ2/σ1= 1, 1.6 or 3.2, and from circular bivariate symmetrical leptokurtic populations with zero means, equal variances and β2− 3 = 3.2 or 6.2. Results suggest that the null distribution ofT2for pairs of bivariate normal samples withN1=N2≥ 10 is rather robust against variance inequality but that this robustness does not extend to disparate sample sizes, and that upper tail frequencies of the distribution of bivariateT2forN1,N2≥ 10 are not substantially affected by moderate degrees of symmetrical leptokurtosis. In simulations of sampling from circular normal populations with scale parameters in the ratio of 1.6:1 and 3.2:1 respectively, the proportion ofMexceeding the null 5% point ranged from 9% and 49% forN1=N2= 5 to 60% and 100% forN1=N2= 20. In simulations of homoscedastic leptokurtic sampling, the proportions ofMexceeding the null normal 5% point forN1=N2= 5 and forN1=N2= 20 were 8% and 17% for β2− 3 = 3.2, and 22% and 42% for β2− 3 = 6.2.