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SPRING 2000

VOL. 51, Nol 1

VIRGINIA JOURNAL OF SCIENCE

OFFICIAL PUBLICATION OF THE VIRGINU ACADEMY OF SCIENCE

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VIRGINIA JOURNAL OF SCIENCE

OFFICIAL PUBLICATION OF THE VIRGINIA ACADEMY OF SCIENCE

VoL 51 No. 1 SPRING, 2000

TABLE OF CONTENTS PAGE

ARTICLES

The Fermat Point and The Steiner Problem. J.N. Boyd and P.N. Raychowdhury. 3

Comparison of Larval Myomere Counts Among Species of Nocomis in Virginia (Actinopterygii: Cyprinidae). Terre D. Zorman and Eugene G. Maurakis. 17

Feeding Habits of Young-of-Year Striped Bass, Morone saxatilis, and White Perch, Morone americana, in lower James River, VA.

Paul J. Rudershausen and Joseph G. Loesch. 23

Binding of Pb and Zn to Aluminium Oxide and Proton Stoichiometiy. Wing H. Leung md A. Kimaro. 39

JEFFRESS RESEARCH GRANT AWARDS 51

NECROLOGY - Blanton Mercer Bruner 58

NECROLOGY - In ing Gordon Foster 60

1 MAY 03 201

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Virginia Journal of Science Volume 51, Number 1 Spring 2000

The Fermat Point and The Steiner Problem

J.N. Boyd and P.N. Raychowdhury,

Department of Mathematical Sciences, Virginia Commonwealth University, Richmond, Virginia 23284-2014

ABSTRACT

We revisit the convex coordinates of the Fermat point of a triangle. We have already computed these convex coordinates in a general setting. In this note, we obtain the coordinates in the context of the Steiner problem. Thereafter, we pursue calculations suggested by the problem.

INTRODUCTION

We came upon a lovely problem in Strang's outstanding calculus text (Strang, 1991). The problem, to which the author attaches the name of the great Swiss geometer J. Steiner, may be stated in the following manner:

Find the point F on the perimeter or in the interior of AF, F2F3 so that the sum of the distances from F to the three vertices of the triangle has its least possible value.

The problem is to minimize + ^2 + <5^3 where d2, d^ are the distances from F

to Fj, F2, F3, respectively. The geometry of the problem is suggested by Figure 1.

We shall eventually show that the point Ffor which di + d2+ d^ is a minimum is located in the interior of AFj F2F3 in such a way that Z F1FF2 = / F2FF3 = / F3FF, = 120° provided that the largest of the three angles of the triangle (Z Fj, / F2, / F3) has a measure less than 120°. If one of these three angles equals or exceeds 120°, then the point F is located at the vertex of that angle. We shall confine our discussion and calculations to triangles in which each angle is less than 120° .

Now let us suppose that three equilateral triangles have been constructed in the exterior of A F, F2F3 with each equilateral triangle having its base upon one of the sides of AV1V2F3 as shown in Figure 2. Then the three lines determined by the three vertices of AFJF2F3 and the corresponding remote vertex of the opposite equilateral triangle are concurrent. The point of concurrence is the equiangular point of AFj F2F3 and is called the Fermat point. These matters have been demonstrated in elegant fashion by Posamentier (Posamentier, 1984), and we have found the convex coordinates of this point (Boyd, Lees, and Raychowdhury, 1990).

When each angle of AF1F2F3 is less than 120° , the Fermat point is also the point F which minimizes di+ d2 + d-^. In this present note, we take advantage of the solution of the Steiner problem to provide an alternative method for computing the convex coordinates of the Fermat point.

THE CONVEX COORDINATES

We shall postpone the solution of the Steiner problem until the last section of our paper and, for the present, assume that we have in hand the distances d^, d2, d^ from

4

VIRGINIA JOURNAL OF SCIENCE

V'l

FIGURE 1. PointFin AF.FjKj .

FIGURE 2. The Equiangular Point is the Femiat Point,

Fto Fj, F2, F3, respectively. These are the distances which minimize 4+ + <^3 for

all points of AFJF2F3 when all angles of AF1F2F3 are less than 120®.

The convex eoordinates of a point P in the closed triangular region V^V2V-^ are three nonnegative numbers aj, aj, (X3, such that ai + a2 + a3 = 1. We say that is the convex coordinate of P with respect to Fj for / = 1, 2, 3. The convex coordinates of Fj are 1, aj = 0 for j ^ i.

FERMAT POINT AND THE STEINER PROBLEM

5

FIGURE 3. in the Cartesian Plane.

Let us suppose that the plane of ^¥^¥21^3 is the Cartesian plane and that the coordinates of ¥^ are (xj, y^). Let us denote the Cartesian coordinates of P by (x, Then the property of convex coordinates which will advance our calculations is that

X = + 6^2X2 +

y= a^y. + a^yj + a.y.

(1)

Thus the Cartesian coordinates of P may be regarded as a convex combination of the Cartesian coordinates of the vertices (Boyd and Raychowdhuiy, 1987).

Let us choose the Cartesian coordinates of F and F, to be (0, 0) and (0, d^), respectively. Then the Cartesian coordinates of Fj are {- d2 cos 30° , - d2sin 30° ) =

( - <^2 V? /2, - d2 /2) and the Cartesian coordinates of F3 are (d^ cos 30°, -dj^in 30°)

= {d^F iy-dJ2) as shown in figure 3.

Equations 1 and the fact that convex coordinates sum to unity yield three linear equations for the convex coordinates of the Fermat point F :

-(c/2 V3 / 2)a2 + (^^3 V3 / 2)0:3 = 0

~ {d2 / 2)a2 - {d^ / 2)a3 == 0 > .

6

VIRGINIA JOURNAL OF SCIENCE

Application of Cramer's rule quickly yields

_ ^2^3 _

d^d2 + d^dj, + d2dj, dxd2

d^d2 + ^1^3 + d2d^

dyd2

3na aj = -73 - 73 - 333 ,

d\d2 + ^1^/3 + <^2^3

(2)

THE ISOSCELES TRIANGLE

Let us denote the length of the side opposite to Fj in AViVjV^ by , / = 1, 2, 3. Then let AFj FjFj be isosceles with ^2 == ^3 = ^ and / Fj < 120®. It should be clear that ZFF2F3 = ZFl 3F2 = 30® and that symmetry implies that dj and must have the same value which we denote by d. Figure 4 indicates the geometry for the isosceles triangle.

We see that (1^ = 2d cos 30°= d V? and d=^i/ V? . Applying the law of cosines

to AVjV2F, we obtain = cf + df - 2d d^ cos 120®. Therefore,

dl -dd^+{d^ -f) = 0

and

^ -d + ^ld^ -4id^ -f)

~ 2

We take the positive sign in applying the quadratic formula since cf-f<0 and d^ > 0. Substituting d= into the last equation, we find that

Substituting d^, dj ~ d, and d^ = d into Equation 2 yields the convex coordinates. As a check upon our work, we let = ^2 ^3 1- Then the triangle becomes

1

equilateral with d^ = dj

and = % expected.

As a further example, let ^, = 1 and ^2 ^3 = 2. Then di = (3 Vs -1)/ (2 Vs ) «

1.6478 and d2 = d^ = 0.5882. It follows from Equation 2 that aj « 0.1514, a 2 » 0.4243, and ^ 0.4243 for Fermat point Fin the triangle defined by the lengths

FERMAT POINT AND THE STEINER PROBLEM

7

of the three sides as listed.

THE SCALENE TRIANGLE

It would seem much more natural to begin the investigation of a triangle with known sides rather than with given values of d^ We shall now show how to

compute di , dj , d^ when AF 1F2F3 is scalene. That is, we begin our computation under the assumption that ^2 ^ ^ 3

We apply the law of cosines to subtriangles V2FV1, V^FV^ and VyFV2eLS shown in Figure 5 to obtain the following equations:

-d\ ^d2d^.

(3.1)

i\ -d\ +dl

(3.2)

and

£\ -di ^-d\ ^rd^d2-

(3.3)

Equations 3 . 1 , 3 .2, and 3.3 possess pleasing symmetry and, as quadratic equations in ^/i , , d^, are solvable in terms of square roots when the given sides satisfy the

8

VIRGINIA JOURNAL OF SCIENCE

V,

FIGURE 5. Tlie Scalene Triangle.

triangle relations 3 < i + 2

Multiplying Equations 3. 1, 3.2, and 3.3 by - dj, - d^, and dj-di, respectively, yields three other equations:

^2^dy—d^) = dY ~ dy ,

and i,\ (d2 ~di) = dl -d^ .

Adding these three equations and performing algebraic simplification yield a fourth equation which we will find useful:

{t\ - l\)dy + (el - £\)d2 + (ii -t\)d,) = 0 (3,4)

For the sake of convenience, we adopt the notation

and i ““^2

>

(4)

FERMAT POmX AND THE STEINER PROBLEM

9

Then Equation 3.4 may be written as

A^di + ^2^2 ^3^3 ^

which implies that

f Ajdi+A2d2^

= Z

K Aj J

We now substitute this expression for c/^ into Equation 3.1. We then perform straightforward algebraic operations to obtain

^3^^ (^3 ^2 "" ^2^3)^2 A-idi + (2^j7l2^1 ■" ^1^3^l)^2

We next rewrite Equation 3.3 as

0 = 4' +^2+ (4' -^3)

and note that d\ - <^ .

Then

^ yj 4(1 -Sdi -4

2

Substitution of this expression for into Equation 5 yields a most complicated equation in the single variable d^.

{A1 -2A^A2+A^ A; )4 ^4(1-34 =

(-4' + 2A^ + 2A^ - 2A1A2 - 2A2A2)dl + (24'^'3 - 2.43'£^ ) . We simplify this equation by letting

p= 4' -24^2 + 4^3,

Q = -A^ + 2AI + 2AI - 2A^A2 - 2A2A2 + A^A^, ' and R = 2A^i\ - 2AI - 2Alt\.

With the substitutions of P, Q, and R into Equation 6, we obtain

Pd2^4(\-3dl =Qdl +R.

(6)

(7)

Squaring both sides of this equation, followed by algebraic manipulation, yields

(3P' + e' )d2 + (2QR -4(\P^ )dl + i?' = 0 .

Application of the quadratic formula then yields

10

VIRGINIA JOURNAL OF SCIENCE

d

2

2

A^\P^ -2QR±^{2QR-Ai\P^f -4(3P^ +Q^)R^ 2(3P^ +0^)

Next we wrote a simple computer program in Quick BASIC to evaluate . It was a simple matter to enter ^ j , £ 2 ’> ^ 3 ^nd then to computed 1, vlj, ^3, P, Q, and

R from Equations 4 and 7. We evaluated d^ (-) by taking the " " sign of the ’’ ±

combinatioalf t/2 (“) turned out to be negative, it was rejected; and we computed d^i^)

for the "+" sign. We took the positive square root of <^2 (+) to obtain d2. Our

program then turned to Equation 3. 1 to find Rewriting Equation 3.1 as

d^ +^2^3 + (^2

we noted that d^ - t\< 0 . We were immediately able to conclude that

2

We then substituted our values for d22ind d^ into Equation 3.4 to obtain d^.

If <^2 (~) positive, we then had to decide between <^2 (™) ^2 (+)

the correct value of To do this, we used d2 (-) to find possible values of d^ and d^. We then checked d2 (-) , <^3 and d^ in the three triangle inequalities

^2 (“) ^3 ^ ^ 1 ? <^3 ^ ^ 2 (”) ^ ^ 3

If all three inequalities were satisfied, then we had found d^, (^2, and d^.

2 2

Ifall three inequalities were not satisfied, we substituted ^2 (+) (”)

repeated the calculations to find the correct values of d^, dj, and d^.

In Table 1, we display the results of five runs of our program. Each triple d^, d2, and dy may be shown to satisfy Equations 3.1, 3.2, and 3.3.

FERMAT POINT AND THE STEINER PROBLEM 1 1

TABLE 1. Five Solutions for and d^.

^1

^2

^3

di

d.

d.

3

4

5

3.388521

2.354003

1.023908

4

7

8

6.083282

2.978962

1.567617

5

6

9

5.690160

4.685959

5.778656

7

9

10

6.816284

4.663640

3.385508

8

11

14

9.573575

6.494169

2.44232

As a last calculation, we return to Equation 2 for the convex coordinates of the Fermat point in the right triangle with ^j = 3, ^4 = 4, ^3 = 5. We find that

ai = 0.173947, 6^2 = 0.250392, and ^3 = 0.575661.

AN INEQUALITY

In the course of our computations, we discovered an inequality which was not obvious to us when we first wrote Equations 3. 1, 3.2, and 3.3 which hold true for all triangles whether isosceles or scalene. In this section, we shall derive that inequality. Adding the three equations to which we have just referred, we see that

+ ^2 ”*~ ^3 4" ^2^3 ^1^3

Dividing both sides above b}' 2, we obtain

H-^2 +^3 ~ ^ ^ j ^2^3 ^ ^\^3

(^1 +<^2 +^3)^ “f (^1^2 +^2^3 +^1<^3)

Since dj ,and d^ are positive, we may conclude that

2

3

<di+ d2 + .

2

(8)

12

VIRGINIA JOURNAL OF SCIENCE

We may also write that

d}

+ d,d

ri

9 9

+di

2 2 3

d}^dl

+^1^3

'> d-^ + 2^j^2 ^d^d:^ + 2dyd^ ^3 *

Therefore,

d 2 “^^3 -^-^1 + ^ 2 "^^3

Taken together, inequalities 8 and 9 imply that

(9)

+d, +d, <^i]+£i+ii

THE STEINER PROBLEM

By solution of the Steiner problem, we mean a demonstration that each of the angles, LjFF2, VjFV^ and V^FV^ (see Figure 5) has measure 120°. Posamentier gives an elegant geometrical argument that, if and only if di + d2 + d^ is a minimum, then the three angles around point F are congruent. The drawback to such a proof is that it is unlikely that even a very good mathematician who does not teach Euclidean geometry would know how to start on his own. Therefore, we shall sketch Strang's solution in which an exercise of ingenuity is not required at the start but is postponed until later. Posamentier presents the result as one of a sequence of theorems and does not associate Steiner’s name with it.

Let the Cartesian coordinates of L,, F2, V3 be (xj, yi), (X2, y2), (^3,73), respectively, and let those of point P in the interior of AV^V2V^ be (x, y). The distance from P to F, is

di = ^(x-x,)\(y-y,y

for / = 1, 2, 3. We seek the point P having coordinates which minimize

g(x,y) = dj+d2+d3.

FERMAT POINT AND THE STEINER PROBLEM 13

To determine these critical coordinates, we let Vg(x, = V 4 + V e/j + V c/3= 0 . Since df =(x - ~ yiY ^ may write that

2d^ - 2(x“ x^ )i+ 2{y- where x,j) are unit vectors in the x,

^-directions, respectively. Therefore

{x-x,)x + {y-y,)y

' d,

and I 1 = 1. Thus each gradient V i - 1, 2, 3, is a unit vector directed from

vertex Fj toward point P. Denoting these unit vectors by , we write the gradient of g(x,y) as

^s{x, y) = tii +U2 +M3 =0.

We also note that the discontinuities of the gradient occur at vertices Fi, Fj, and F3 Since the sum of the three unit vectors is the zero vector, it follows that the angle between each two of these unit vectors must be 120°.

The point P of the interior of is precisely F, the equiangular point with

which we have been concerned all along. The condition that Vg(x, y)= 0 at F = F is necessary if d^ is to be a minimum value, but it is not sufficient. That the

gradient of g(x, y) is the zero vector at F = F implies only that we have found a

stationary point for g(x, Since the geometry presented by Posamentier makes it clear that di + d2 + d^ takes its least value at F, we dispense with the tests with the higher partial derivatives of g(x, y) which may be employed to complete Strang's argument. Strang also fails to test the minimum.

If one of the angles of AFi F2F3 is permitted to equal or exceed 120° the vertex of the largest angle is the point of the closed triangular region for which + <^2+ d^ is a minimum. (Recall that Vg(x, y) does not exist at the vertices.)

A PHYSICAL SOLUTION OF STEINER'S PROBLEM Let us draw or imagine drawing a triangle of convenient size upon the smooth plane surface of a plyboard lamina. Tthe triangle will represent the general triangle of interest in our work. That is, each angle of the triangle should be less than 120° .

14

VIRGINIA JOURNAL OF SCIENCE

FIGURE 6. The Physical Interpretation of the Steiner Problem.

Then we need to drill a small hole through the lamina at each vertex.

Next, we take three strings and tie their ends together (one end from each string) in a single knot. We then thread the free ends of the strings through the three holes (one string through each hole). We obtain three masses of equal weight and attach one to the free end of each string (Figure 6).

If we turn the lamina so that the smooth surface and triangle are upward and horizontal, the system of weights will find its equilibrium configuration with the knot at the equiangular point of the triangle.

The reason is that the tension in each string is equal to the weight that it supports. Since the weights are the same, the three tensions are equal in magnitude. Since the

vector sum of the tensions at the knot must be 0 under equilibrium conditions, the angles between the strings must be 120°. (The reasoning is precisely the same as that

which justified jp) = /q +U2 + W3 =0 in the preceding section. Of course,

we have ignored any frictional effects exerted on the strings as they pass through their holes. We have already assumed the surface of the lamina to be smooth.)

Now, displace the knot horizontally. When the system was in equilibrium with the knot at the equiangular point of the triangle, the value of the gravitational potential energy of the system was minimum, subject to the geometrical constraints of the lengths of the strings. Moving the knot must increase the gravitational potential energy of the system. Thus the center of mass of the three weights below the lamina must rise. When that occurs, the total length of string below the lamina must have been shortened. Since our strings do not stretch, the new position of the knot requires that there now be a greater total length of string from the knot to the three vertices

FERMAT POINT AND THE STEINER PROBLEM 15

than before the knot was moved.

We may conclude that the equiangular point is the point which solves the Steiner problem.

LITERATURE CITED

Boyd, J.N., Lees, J.J., and P.N. Raychowdhuiy. 1990. The Convex Coordinates of the Fermat Point. Virginia Journal of Science 41:487-491.

Boyd, J.N. and P.N. Raychowdhury. 1987. Convex Coordinates, Probabilities, and the Superposition of States. The College Mathematics Journal 18:186-194. Posamentier, A.S. 1984. Excursions in Advanced Euclidean Geometry. Addison-Wesley, Reading, MA.

Strang, G. 1991. Calculus. Wellesley-Cambridge Press, Wellesley, MA.

VIRGINIA JOURNAL OF SCIENCE

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Virginia Journal of Science

Volume 51, Number 1

Spring 2000

Comparison of Larval Myomere Counts Among Species of Nocomis in Virginia (Actinopterygii: Cyprinidae)

Terre D. Green and Eugene G. Maurakis, Science Museum of Virginia, 2500 West Broad Street, Richmond, VA 23220, and Department of Biology, University of Richmond, Richmond, VA

23173

ABSTRACT

Larval myomere counts of Nocomis platyrhynchus were made using a dis¬ secting light microscope equipped with polarizing filters, and then compared to those of the three other species of Nocomis (Nocomis leptocephalus, Nocomis micropogon, and Nocomis raneyi) found in Virginia. Average preanal myomere counts for N. platyrhynchus (26,9) were significantly different from those of the other tliree species (V raneyi =28.7; N. micropo¬ gon =26.0; and N. leptocephalus =25.9). This is especially important as larvae of N. leptocephalus, the only otlier species of Nocomis syntopic with N. platyrhynchus in the upper New River drainage, can now be distinguished from those of N. platyrhynchus. Larvae of N. raneyi also can be distinguished from those of otlier species of Nocomis in Virginia based on preanal myomere counts. However, larvae of N. platyrhynchus and N. raneyi cannot be distin¬ guished from each otlier based on total myomeres (42.0 versus 4 1 .7). Larvae of N. platyrhynchus can be distinguished from those of three of its nest associates (Lythrurus ardens, Notropis rubellus, and Phoxinus oreas), but not from Campostoma anomalum and Luxilus chrysocephalus using myomere counts.

INTRODUCTION

Nocomis (Cyprinidae) is composed of seven species in three species-groups: Nocomis biguttatus witli Nocomis asper and Nocomis effusus; Nocomis leptocephalus', and Nocomis micropogon with Nocomis platyrhynchus and Nocomis raneyi (Lachner and Jenkins, 1971; Maurakis et al., 1991). Of these, four species (N. leptocephalus, N micropogon, N platyrhynchus and N. raneyi), representing two species-groups, are found in Virginia. Nocomis leptocephalus occurs in the New, Potomac, Rappahan¬ nock, York, James and Roanoke River drainages. Nocomis micropogon is distributed in tlie Potomac, Rappaliannock, York and James River drainages. Nocomis platyrhyn¬ chus is endemic to the New River drainage, and N. raneyi occurs in the James and Roanoke River drainages in Virginia (Jenkins and Burkhead, 1994) (Table 1).

Total, preanal, and postanal myomeres of larval N leptocephalus, N micropogon, and N. raneyi have been counted and compared by Maurakis et al. (1992). Currently, however, there is no information on the numbers of myomeres of larvae of N. platyrhynchus. The objective of tliis study is to present information on myomere counts of larval N platyrhynchus, and compare them to counts of the other three species of Nocomis in Virginia, as well as nest associates of N. platyrhynchus.

18

VIRGINIA JOURNAL OF SCIENCE

TABLE 1. Rivers of Virginia containing Vocow/j leptocephalus, Nocomis micropogon, Nocomis platy- rhynchus, and Nocomis raneyi.

Species

River

New

Potomac

Rappahannock York

James

Roanoke

leptocephalus

X

X

X X

X

X

micropogon

X

X X

X

platyrhynchus

X

raneyi

X

X

MATERIALS EXAMINED

Tlie state, drainage, larvae (L), eggs (E), collection number (ANSP, Academy of Natural Sciences of Pliiladelphia; and EGM, Eugene G. Maurakis), locality, and collection date for K leptocephalus, N. micropogon, N. platyrhynchus, and N. raneyi are:

Nocomis leptocephalus. Georgia: Savannah, (L), ANSP 140977, Columbia Co., Reed Cr., Rt. 28, 3.4 km N of jet. with Co. Rt. 26 near Martinez, 26 June 1976. North Carolina: Savannah, (L), EGM-NC-210, Jackson Co., Horsepasture R., U.S. Rt 64, 2.2 km NE of Cashiers, 10 June 1988. Virginia: Roanoke, (L), ANSP 134421, Montgomery Co., Roanoke R. at Elliston, 13 June 1975.

Nocomis micropogon. Virginia: Potomac, (L), EGM-VA-254, Loudoun Co., Ca- toctin Dr., Co. Rt 633, 0.2 km from Co. Rt. 665 jet near Lovettsville, 25 May 1990.

Nocomis platyrhynchus. Virginia: New, (L), EGM-VA-416, Montgomery -Floyd Co. line, Little R., 1 km upstream of State Rt. 8 bridge on dirt road, 16 May 1998. New, (E), EGM-VA-417, Montogomeiy Co., Little R., Co. Rt. 693 and 613 junction, about 8 km W of Riner, 16 May 1998.

Nocomis raneyi. Virginia: James, (L), EGM-VA-260, Craig Co., Jolms Cr., Co. Rt 632 at Maggie, 13 June 1990.

MATERIALS AND METHODS

Naturally-spawned eggs were collected with an aquarium dipnet from spawning areas of active nests of each of the four Nocomis species. Eggs were transported in tagged plastic jars to the laboratory where they were reared at room temperature (22®C) to fiilly scaled juveniles. Larvae were sampled at each of tliree larval stages (protolar- val, meso larval, and metalarval) following terminology in Fuiman (1982), and pre¬ served in Bouin’s fixative.

Preanal and postanal myomeres of larvae of each of the four species of Nocomis were counted using a dissecting light microscope equipped with polarizing filters. A vertical line was drawn at the posterior end of the anus, and any myomere tliat intersected tliis line was counted as preanal according to methods in Fuiman (1982). Total myomeres were calculated by adding preanal and postanal myomeres of each specimen. Eggs of Nocomis platyrhynchus were measured using a metric rale under the microscope.

Differences in average numbers of each of preanal, postanal, and total myomeres of larvae among the four species were determined with a General Linear Model and Duncan’s Multiple Range Test (SAS, 1985).

LARVAL MYOMERE COUNTS OF Nocomis

19

TABLE 2. Average (range) preanal, postanal, and total myomeres of Nocomis leptocephalus, Nocomis

micropogon, Nocomis raneyi and Nocomis platyrhynchus. Underscored means do not differ significantly

(P-0-05).

Myomere Mean (range) F value Pr > F

leptocephalus

(n = 21)

micropogon (n = 24)

platyrhynchus (n = 53)

raneyi (n = 25

Preanal

25J6 (24-28)

26.04 ,(24-28)

26.88 (25-29)

28.72(27-31)

42.69

0.0001

Postanal

12.24(1 M3)

M2^0 (11-14)

15.09(14-18)

13.00(11-15)

78.52

0.0001

Total

38.10(36-41)

38.54 (36-41)

41.96(40-45)

41.72(39-44)

73.97

0.0001

TABLE 3. Preanal myomere modes and means of Nocomis leptocephalus, Nocomis micropogon, Nocomis

platyrhynchus, and Nocomis raneyi.

Species Mode _ _ ^

24 25 26 27 28 29 30 31

leptocephalus 2 micropogon 2 platyrhynchus raneyi

5 9 4

4 10 7

2 17 22

4

1

1

10 2

6 9 5

25.86

26.04

26.88

28.72

When myomere counts were unavailable (e.g. Lythrurus ardens), an adjusted vertebral count (total minus one vertebrae) was used as a prediction of myomeres using metliods of Fuiman ( 1 982). Vertebral counts of Phoxinus areas were determined from radiograplis and were not adjusted as our counts of both myomeres and vertebrae included final elements. Specimens were exposed at 30kV and 5mA for 45 seconds.

RESULTS

Average total myomere counts of N. leptocephalus ^ =38. 10) and N. micropogon (x = 38.54) are significantly lower than those for N. raneyi (i = 41.72) and N. platyrhynchus (x =41.96) (Table 2). Preanal as well as postanal counts of N. raneyi (x = 28.72, 13.00) and N. platyrhynchus ^ = 26.88, 15.09) differ significantly from those of N. leptocephalus (x = 25.86, 12.24) and N. micropogon (x = 26.04, 12.50) (Table 2). The modal preanal and total myomere counts for all four species are consistent with their respective mean values (Tables 3 and 4). Eggs of N. platyrhynchus averaged 2.1 mm in diameter, ranging = 2.0-2.2 mm.

20

VIRGINIA JOURNAL OF SCIENCE

TABLE 4. Total myomere modes and means of Nocomis leptocephalus, Nocomis tnicropogon, Nocomis platyrhynchus, and Nocomis raneyi.

Species

Mode

X

36

37

38

39

40

41

42

43

44

45

leptocephalus

4

2

7

5

2

1

.

.

.

.

38.10

micropogon

1

5

5

8

3

2

-

-

-

-

38.54

platyrhynchus

-

-

-

-

7

11

18

13

2

2

41.96

raneyi

"

"

"

1

3

7

6

7

1

41.72

TABLE 5. Actual and predicted preanal, postanal, and total myomeres of larvae of nest associates of

Nocomis platyrhynchus.

Preanal

Postanal

Total

Campostoma anomalum

26-29

11-14

37-43

Luxilus chrysocephalus

26-27

12-14

38-41

Lythrurus ardens *

18-20

17-20

36-40

Notropis rubellus

19-23

15-18

34-41

Phoxinus oreas *

21-23

17-19

38-41

X

22.1

17.4

39.5

* Myomere counts of L. ardens predicted from vertebral counts of Snelson (1972) using methods in Fuiman (1982), and myomeres predicted for P. oreas from our vertebral radiographs.

DISCUSSION

Nocomis platyrhynchus can be distinguished from other species of Nocomis in Virginia based on preanal, postanal, and total myomere counts with one exception. Tliis is especially significant as N leptocephalus is the only Nocomis species tliat occurs witli N. platyrhynchus in tlie New River. Nocomis platyrhynchus cannot be separated from N. raneyi with total myomere counts. Tliat larvae of N. raneyi also can be distinguished from those of N. leptocephalus and N. micropogon based on preanal and postanal myomere counts is consistent with results reported by Maurakis et al. (1992). Myomere counts of larval N platyrhynchus are consistent with those of vertebrae (x = 40.6, range = 39-42, mode = 41) in adults reported by Laclmer and Jenkins (1971).

Larvae of N. platyrhynchus can be distinguished from those of some of its nest associates (i.e., species that congregate and may spawn over a nest but do not contribute to its construction) by myomere counts (Table 5). Lobb and Orth (1988) and Maurakis (1999) reported Campostoma anomalum, Luxilus chrysocephalus, Lythrurus ardens, Notropis rubellus, and Phoxinus oreas as nest associates of N. platyrhynchus. Larvae of N. platyrhynchus can be separated easily from those of L. ardens based on preanal myomeres as N. platyrhynchus has 25 to 29, and Snelson ( 1 972) reported precaudal vertebra counts forL. ardens to be 18-20. Notropis rubellus can be distinguished from N. platyrhynchus based on preanal myomeres (range =19-23,

LARVAL MYOMERE COUNTS OF Nocontis

21

Fuiman and Heufelder, 1982). Larvae of P. oreas may be differentiated from N. platyrhynchus by preaml myomeres adult P. oreas have precaudal vertebra counts of 21-23. Nocomis platyrhynchus cannot be distinguished from C anomalum on tlie basis of myomere counts. However, they can be distinguished by egg diameter {N. platyrhynchus, i = 2.1 mm; range = 2.0-2.2; n = 10; C anomalum, range - 2.3-2.4 mm, Auer, 1982), as well as by their spawning location. Nocomis platyrhynchus spawi^ in a trough on tic upstream slope of the nest whereas C. anomalum spawns in pits on this slope. Larvae of L chrysocephalus cannot be distinguislcd from those of M platyrhynchus based on numbere of preanal, postanal and total myomeres of Fuiman and Heufelder (1982) or egg diameter (range = 2.0-2.3 mm) reported by Auer (1982).

Maurakis et al. (1992) indicated that the increased resolutions of scanning electron microscopy and compound light microscopy are superior to that of dissecting light microscopy for identification and enumeration of actual numbers of myomeres in larvae. Tliey proposed that myomere counts made with dissecting light microscopy alone are not accurate where the actual number of myomeres is required. However, in tliis study, due to the great divergence in preanal, but particularly total myomere means and modes in N. leptocephalus and N platyrhynchus, dissecting light microscopy is adequate for distinguishing larvae of tlie two species in the New River drainage.

ACKNOWLEDGEMENTS

This study was funded in part by grants awarded to T. D. Zorman by the Under¬ graduate Research Committee of University of Richmond, to E. G. Maurakis by the Small Research Fund Committee of the Virginia Academy of Science, and the Science Museum of Virginia. We sincerely thank Frank Schwartz, University of North Carolina, for making radiograplis of Phoxinus oreas.

LITERATURE CfTED

Auer, N. A. (ed.). 1982. Identification of larval fishes of the Great Lakes basin with emphasis on tlie Lake Micliigan drainage. Great Lakes Fishery Commission, Ann Arbor, MI 48105. Special Pub. 82-3:744 pp.

Fuiman, L. A. 1982. Correspondence of myomeres and vertebrae and their natural variability during tlie first year of life in yellow perch, pp. 56-59. In C. F. Bryan, J. V. Conner, and F. M. Tmesdale (eds.). The 5^ Ann. Larval Fish Conf. La. Coop. Fish. Res. Unit and School of Forestiy and Wildl. Mgt. La. St Univ., Baton Rouge, LA.

Fuiman, L. A. and G. R. Heufelder. 1982. Family Cyprinidae, carps and miimows. pp. 242-249, 270-276. In N. A. Auer (ed.). 1982. Identification of larval fishes of the Great Lakes basin with emphasis on the Lake Michigan drainage. Great Lakes Fishery Commission, Ann Aibor, MI 48105. Special Pub. 82-3:744 pp. Jenkins, R. E. and N. M. Buridiead. 1994. Freshwater Fishes of Virginia. Amer. Fish. Soc. Betliesda, MD. 1079 p.

Lacliner, E. A. and R. E. Jenkins. 1971. Systematics, distribution, and evolution of the chub genus Nocomis Girard (Pisces, Cyprinidae) of eastern United States, with descriptions of new species. Smitlis. Contr. ZooL 85: 1-97.

Lobb, M. D. and D. J. Ortli. 1988. Microhabitat use by the Bigmouth chub Nocomis platyrhynchus in the New River, West Virginia. Am. Midi. Nat 120(l):32-40.

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VIRGINIA JOURNAL OF SCIENCE

Maurakis, E. G., W. S. Woolcott, and M. H. Sabaj. 1991. Reproductive-behavioral phylogenetics of Nocomis species-groups. Am. Midi. Nat. 126: 103-1 10. Maur^is, E. G., W. S. Woolcott, G. P. Ra(ice, and W. R. McGuire. 1992. Myomere counts in larvae of three species of Nocomis (Pisces: Cyprinidae).

SAS Institute, Inc. 1985. SAS User’s Guide: statistics. Version 5. SAS Institute, Caiy, NC. 956 p.

Snelson, F. F., Jr. 1972. Systematics of the subgenus Lythrurus, genus Notropis

(Pisces: Cyprinidae). Bull. Florida State Mus. 17(l):l-92.

Virginia Journal of Science Volume 51, Number 1

Spring 2000

Feeding Habits of Young-of-Year Striped Bass,

Morone saxatilis, and White Perch, Morone americana,

in lower James River, VA

Paul J. Rudershausen, Tarpon Bay Environmental Laboratory, 900 A Tarpon Bay Road, Sanibel, FL 33957 and Dr. Joseph G. Loesch, Virginia Institute