In astrophysics the polytropic law with index n is commonly used as a means of imposing a simple and ordered physical structure on a gaseous (or smoothed discrete) system. In many instances it would be preferable to be able to introduce a polytropic density variation analytically into the basic theory rather than numerically at the computational phase. It is perhaps unfortunate that the three well known classical analytical E type solutions of the Lane-Emden equation for n = 0, 1 and 5 all have some constraining physical features; specifically, the polytrope n = 0 has uniform density and hence arbitrary radius, when n = 1 the mass and radius are independent of each other and the solution cannot be transformed homologically, and because the first zero ξ1 = ∞ for n = 5 the corresponding polytropic model has infinite extent and central condensation. In contrast, and unlike most stars, the two finite radius models have central condensations which ~ 1.