Egg Volume

Abstract

PRESTON (1974, Auk 91: 132) has given equations for calculating the volume of an egg, assuming that the shape of an egg can be described by the revolution about its long axis of an oval figure whose parametric equations are y=bsinO (1) x = a cos 0 (1 + c• sin 0 + c2sin 2 0), (2) where ct and c2 are coefficients representing the departure of the oval from an ellipse. In particular, ct represents a departure from symmetry, being zero for a symmetric egg. Preston's analysis was in two parts. First he expressed the volume, V, in terms. of the length, L, and the breadth, B, measured half way between the poles of the egg. Then he found a relation between the easily measured maximum breadth B .... and B. A further step, which would be most immediately useful in practice, would be to combine the two parts of this analysis and to give an expression for V directly in terms of the measurable quantities L and Bmax. Preston's formula for V was developed to first order in c• and c2 (To this order it is independent of ct.) His formula for B .... /B was developed to second order in ct and c2 (To this order it is independent of c2.) In order to express V directly in terms of L and Bmax, it is necessary that both parts of Preston's analysis should be carried to the same order. The analysis here is carried to the second order. For the volume in terms of the length and equatorial breadth we have, to second order in c• and c2: ,r L B 2 V-• (1 + 2•c2 + %c• 2 + a/•a c22) , (3) in agreement, to first order, with Preston's expression. In retracing Preston's derivation of his expression for B ..... lB, I noted some small errors. Preston's equation (8) should read 362 sin a 0m + 2C• sin 2 0m-3-(1-2C2) sin 0,•-c• = 0. (4) For small values of sin 0m, CX and c2, Preston finds sin 0m m cx. This is actually correct to first order and follows from the above equation (4), although it would not follow from Preston's erroneous equation (8). However , in order to derive B•x/B correct to second order, sin 0m must also 576 The Auk 92: 576-580. July 1975