THE VOLUME OF AN EGG F. W. PRESTON IN MEMORIAM, F•A•c•s JosEPIt ASlIBY, 1895-1972 with whom I collected my first birds' eggs June 8, 1906, in central England, shared my first seafowl eggs from North Wales a little later, and with whom I went birding and bird's-nesting for 60 years. RECENTLYI have receivedseveral inquiries as to how one calculatesthe volumeof an egg from its "dimensions,"and by dimensionsis meant the length and the maximum breadth. The answer is that it cannot be done with any real accuracyon the basisof only two measurements.I have shown(Preston 1969) that the "shape,"i.e. the longitudinalcontour, of an eggand its sizecan be describedwith a high order of accuracyby meansof four parameters,length, breadth, asymmetry,and bicone,and that it can- not usuallybe describedwith less. The contourdetermines the volume,and hencevolume cannot be estimatedfrom two measurementsonly. In crosssection an eggis remarkablycircular. It is thereforelegitimate to consideran egg as a "surface of revolution," and this assumptionis alwaysmade. An egglies betweentwo simplegeometrical figures, a cylin- der and a true bicone. In Figure 1A, we showa cylinder of length L (= 2b) and diameter B (----2a). Its volumeis (,r/4) ßLB 2 or 2,rba2 An ellipsoidalegg of length L and diameter B would lie entirely inside the cylinder,touching the centersof both endsand making contactwith the cylindrical surfaceon a circle. In Figure lB, we showa bicone. Its volumeis (,r/12) ßLB 2. Our egg would lie entirely outsidethe surfacesof the bicone,and, if symmetrical,it would passthrough the apicesof the two conesand would touch the bases of the conesall the way round. If our eggis asymmetrical,it would touch the samepoints and placesof the biconein Figure 1C, whosevolume is still (?/12) ßLB 2. We may surmisetherefore that in real eggs,which lie between Figures 1A and 1C, asymmetry(the extent to which one end is biggeror blunter than the other) makeslittle differenceto volume, but biconemakes a great deal, the coefficientof LB 2 varying somewhatbut lying between (,r/4) and (*r/12). The preliminary assumptionof the early writers is that the egg is, to a first approximation,an ellipsoidof revolution. If so, its volume would be (,r/6) ßLB 2. The approximationis sometimesquite good, and taking ßr = (22/7), the formulabecomes 132 The Auk 91: 132-138. January 1974 January 1974] Volume o] an Egg 133 • ELLIPSE // Figure 1. A, a cylinder of length L (• 2b) and diameterB (---- 2a) may be regarded as circumscribingany egg,even a hummingbird's.B, a bicone (two conesbase to base) will lie inside any egg, even a tinamou's,which will circumscribeit. C, the two cones do not needto be identical in height, though they must have the basein common. D, a circle circumscribesan ellipse, touching only at the ends or poles. The "eccentric angle" defines a parameter in terms of which the x and y coordinatesof the ellipse, or of the oval, can be expressed. 11 v =- ßLB 2 (1) 21 a formula usedby someearlier writers. However the approximationis sometimesnot good. The hummingbirds (see figure in Preston 1969) lay blunt-endedeggs halfway betweenthe ellipsoidand the cylinder,so the coefficientof LB 2 is much higher than 11/21, while the grebesand tinamouslay eggsbetween the ellipsoidand 134 F.W. Pa•s•o• [Auk, Vol. 91 the bicone,i.e. they are pointed at both ends like an American or Rugby football, so the coefficientis less than 0r/6) (= 11/21). For a more de- tailed discussionof the inadequaciesof various proposedformulae see Barth (1953). The questiontherefore is, in the formulaV = k LB s, what is the appro- priate value of k? Obviouslythere is no one value that will fit all species. Indeeda singlevalue will not fit all speciesof a singlefamily, and in some clutchesI haveseen it will not fit all eggsof a singleclutch. BelowI givea mathematicaltreatment in moredetail. As in Figure 1D, draw a circle circumscribingthe egg,i.e. touchingthe two endsof the egg. From the center of the circle draw any radius of the circle,making an angle(90 ø - 0) with the long axisof the egg,till it meets the periphery of the circle at a point P• (shown but not marked in Figure 1D). From P• draw a horizontal line to meet the oval or ellipse at point Pa (again, shownbut not marked in Figure 1D). Then the coordinates(x, y) of Pa can be specifiedin termsof the angle 0, which is called the "eccentric angle," (see Preston 1953) and of the length and (equatorial) breadth of the egg. DIGEST The parametric equation of the longitudinal sectionof an egg may be taken (v. Figure 1D) as yx= ab cossin 00(1 + c•sin0+ casin a0 + etc.) } (see Preston 1953) (2) where 0 is the "eccentricangle," a is the semidiameterat the true equator (i.e. halfwaybetween the two endsof the egg), b is the half-lengthof the egg, c• and ca are coefficientsthat vary from egg to egg and have to be found experimentally,and the terms labelled "+ etc." can usually be neglected.c• and ca are usuallyquite small,so that c•a, c2a, and c•cacan be neglected. Slicing the eggparallel to the equator (perpendicularto the long axis of the egg) into small thicknessesdy, gives us various elementsof volume dV = •rx a ß dy and the total volume of the egg is V= f_•r x•dy (3) Ignoring termsthat includenegligible coefficients, we have xa ----aacos a 0 (1 + 2c•sin0 + 2casina 0) January 1974] Volume o! an Egg 135 and of course we have dy--bcos0'd0 So V: •r a2b r/2.cosaO(l+2qsinO+2c2sin20) dO (4) •'-•r/2 The completeintegral from •/2 to +•r/2 of the middle term vanishes (being -cos4 O) and the integral reducesto V = • a2b r/2(cos a 0 + 2c2cosa 0 sin2 O) d 0 (4a) • -•/2 and by writing cosa 0 = cos0( 1 - sin2 0), this integratesto 2 ) (S) If the length of the egg is L (: 2b) and its equatorial (not necessarily maximum)breadth is B (= 2a) this equationtakes the form V=5'LB6 2(2) 1+•c2 (5a) If c• is zerothis reducesto V = (•/6) ßLB 2, the volumeof an ellipsoid of revolution,and it doesnot dependon c• at all, providedwe were justi- fied in assumingc• is comparativelysmall and q2 negligible. c2 can be either positiveor negative. With most speciesand individualparents, c2 is negative,so the volume of the egg is less than the volume of the circum- scribingellipsoid. But •th hummingbirdsand someothers it is positive, and the volume is then more than that of the ellipsoid. EFFECTOF USINGBmax INSTEAD OF Bequatorial inthe parametric equationy:x = ba cossin 0 ( 1 + qsin 0 + c2sin20 } the maximum value of x is obtained when dx/dy (not dy/dx) is zero, or, what is just as good,when dx/d0 = 0. Let 0mbe the value of 0 that makes x a maximum. If cx = c2 ----0, the equationof an ellipse,we get dx --= -a sin 0, and this is zero when 0----0: (6) dO and this is a correct solution. Now let c2 = 0 but let c• be non-zero. Then, rememberingthat sin 0mis assumedsmall and thereforethat cos0m is very near unity, we get 136 F.W. PRESTON [Auk, Vol. 91 -1 +•/1 + 4cxa sin 0m•--- (7) Rememberingthat cx is much less than unity, the squareroot term is very nearly (1 + 2cx2),so that sin 0m= cs very nearly. (7a) If cxand ca are both non-zero, but cos0m is very nearunity (sin 0mbeing small), we get a cubicequation for sin 0 as follows: c2sins 0m•- cssin• 0m+ (1 -- 2c2)sin0m- C1= 0 (8) This can be solvedexactly, but if sin0m is small (it is often lessthan 0.1), we can neglectthe term in sins 0mand get a simplequadratic, whose solu- tion is -(1 - 2c2)+ •/(1 - 2c2)2 + 4%2 sin 0m= (8a) 2c• We may note that when c• and c2 are small (and cs tends to average about0.1 and c2about -0.1) sin 0mtends to be about cs,independently of the valueof c2. (9) This locatesthe positionof the maximumdiameter. The value of that diameter is Bmax= 2Xmax---- 2a COS0n• (1 + cssin0m + c•sin• 0m) = 2a ßX/1 - cs'• ß (1 + cs2 + C2C12) or BmaxB -(1 _ C12• ignoringc2c• 2 as very small,and so, rememberingour previouscomment on the square root term, BmaxB = 1+ c122 (10) For instanceif c½= 0.1, Bm•xexceeds B by about one-half of one per- cent. ERRORSIN ESTIMATINGVOLUME EROM THE Two DIMENSIONS,L and B•nax If an experimentermeasures the lengthand maximumdiameter of an egg and calculatesits volumeby the ellipsoidalformula as V = -• ßLB2m•x 6 January 1974] Volume of an Egg 137 he will overestimatethe volume in the ratio (Braax/B)2, that is (from equation 10) in the ratio (1q-c•-) 2or 1 q- cx2 approximately (11) and if cx= 0.1, the erroris 1%, whileif cx: 0.2 (a valueit reachesin only a few families) the error will be 4%. But this is not the most serious error. A more important error is that due to the biconeterm. Sincethis is usu- ally negative,it will again producean overestimate,of value (2/5)c2. Thus if c2--- 0.1, the overestimateis 4%. In the caseof the Oystercatcher of Preston1953 the error would be 7%. In the caseof the hummingbirdor albatrossof Preston1969 (p. 259), the errorwould be a very substantial underestimate. Worth (1940) considersthat it is closeenough to assume,for any egg, that its volumeis 15%less than that givenby the ellipsoidalformula, and this I cannot concede. It is no use trying to refine the estimate of the volume of an egg by modifying arbitrarily the coefficientk in the expression V = k LB2raax from its ellipsoidalvalue of •-/6.