NOTE D'Amours and Gr^goire: Analytical correction for oversampled Scomber scombrus eggs 



where N(z) = concentration of eggs in number per 

 volume at depth z, and k = rate constant. Upon in- 

 tegration of Eq. 1, the concentration of eggs at depth 

 z is given by 



N(z) = N^e-^, 



(2) 



where Nq = concentration of eggs at the surface. 

 When integrating Eq. 2, the total number of eggs (N^) 

 in number per surface area in this body of water is 

 given by 



Na= f 

 ■J ( 



No 

 N^e-k^dz = — . 

 k 



(3) 



If an oblique plankton tow (Fig. lA) is made through 

 this distribution of eggs with a net of radius a, and a 

 centered depth of a, the total number of eggs collected 

 (Nh) will be equal to 



N. 



(4) 



;Li _ o + a +\/a--{z-ay 



[ ( N(z) dydzdxi 



Va--(z-a) 



/Lj o + a + \/a? - (z-a)' 



I I N(z) dydzdx2, 



■\Jar-(z-aY 



Figure 1 



Path (broken line) of net during an oblique plankton tow: L's 

 represent length components of the tow along horizontal ref- 

 erences X] and X; ; D is the maximum depth of sampling, and 

 e's represent angles between path of net and horizon. In (A), 

 the net is recovered upon reaching the surface; in (B), the 

 net is dragged at the surface along Lp before recovery. 



where Xj is the horizontal distance from the start of the tow and X2 is the horizontal distance from the end of 

 the tow (Fig. lA), and where z and y represent the vertical and horizontal openings of the net, respectively. In- 

 tegrating Eq. 4 over the limits on z and y (as in D'Amours 1988), 



N. 



1 + 



(ka): 



I Noe-k^dxi + I 



[-' -^ 



N„ e-"^ dxzl. 



(5) 



The term ka originates from the slight difference between the position of the geometric center of the net and 

 the position of the center of abundance of the eggs within its opening (D'Amours 1988). Defining, 



tan 0; = — , where D is the maximum depth of the tow, Eq. 5 can be rewritten as 

 Li 



Nh = 



1 + 



(ka)= 



I N« 



e-ktanfliXi (Jxj + 



/, 



M e-ktane2X, dxp 



(6) 



