APPLICATION OF SOME OF CAUCHYS THEOREMS TO ESTIMATION OF SURFACE-AREAS OF LEAVES, NEEDLES AND BRANCHES OF PLANTS, AND LIGHT TRANSMITTANCE

Some simple, general relationships between surface areas of solids and their projections presented by Cauchy have been applied to plants. In the estimation of areas of plant parts such as needles and branches with planimeters, where the parts are placed on a plane for the measurements, surface areas can be obtained from the mean plan areas where the averages are taken for rotation about the long axes of the parts. For any long solid with a convex cross-section, the surface area is pi times the average plan area. The correct surface area is obtained with pine needles which twist about their long axes, and twigs with essentially circular cross-sections. In the field, where parts of plants are free in all dimensions, total surface area index, F, of leaves, needles, branches and stems can be estimated collectively from gap frequencies as deduced from the penetration of probes, rays or the direct beam of the sun. If gap frequencies are determined at a suitable range of probe angles to the vertical, theta, then F = 4-integral 0 pi/2 kappa-(theta)sin theta d-theta, where kappa-(theta) is a suitable transformation of observations of gap frequencies. To a good approximation F = 4(A + B), where A and B are the regression coefficients for a linear fit of observations of kappa-(theta) against theta in radians. If the surface area indices of woody parts are measured or estimated separately, they may be subtracted from the total to give an estimate of the surface area index of foliage.