Observations of 14 dense cores in lines of CS, (CO)-O-18 and NH3 are combined to examine the relations between the Doppler width of a line and the spatial extent of the map of line intensity. The line width increases with map size R as R(p). The deduced width DELTA-upsilon(TOT) of the velocity distribution of the molecule of mean mass has p = 0.20 +/- 0.04, and the deduced width DELTA-upsilon(NT) Of the nonthermal velocity distribution has p = 0.7 +/- 0.1. The values of p for eight cores with embedded stars and for six cores without embedded stars are statistically indistinguishable, but p is better determined for cores without stars. Thus the physical basis of the line width-size relations is part of the initial conditions for star formation, rather than a consequence of the star-core interaction. The cores appear close to virial equilibrium, consequently the variation of number density with distance r from the core density maximum is deduced. Expressed as a single power law, the number density varies as r(-q), with q = 1.6 +/- 0.1, and the nonthermal pressure varies as r(-s), with s = 0.2 +/- 0.1. If the nonthermal pressure is magnetic, the field strength increases with density as B is-proportional-to n0.05. The density variation within the cores is also deduced for a model where the thermal and nonthermal motions imply two corresponding power law components of density.