FISHERY BULLETIN: VOL. 69. NO. 2 



then direct comparisons between species are im- 

 possible, but if they do not exist then a general 

 model can be derived from our data which can 

 be used to make specific comparisons. In the in- 

 terpretation of our data on Trachurus we will 

 consider the alternatives, Case I where all swim- 

 ming functions are related to size on a species- 

 specific basis, and Case II where swimming 

 functions are proportional to the same power of 

 length in different species. 



To evaluate Case I where length coefficients 

 are considered to be species-specific we regressed 

 speed on frequency by the general relationship. 



V = a.L" 



+ 



a^L* * F 



where V is swimming speed in centimeters per 

 second. F is tail beat frequency in beats per sec- 

 ond, L is total length in centimeters and w.L"' 

 is the intercept function and ajL"^ is the slope 

 function for the tail beat frequency-swimming 

 speed relationship. Estimates were obtained by 

 use of Marquardt's Algorithm for fitting non- 

 linear models (Conway, Glass, and Wilcox, 

 1970). For Trachurus the 90% support-plane 

 confidence intervals (Conway et al., 1970) for 

 03 and 04 were 0.72 <«3 <1.82 and 0.72 <a^ 



<1.01 where S, = 1.28 and «, 



0.86. 



To evaluate Case II where length coefficients 

 are the same for all fish we assumed the slope 

 coefficient a^ equaled one. When a^ = 1, a^ = 



0.86 with 90''f confidence limits of 0.79 < «, 

 < 0.91. On the basis of the Trachurus data 

 alone there seems to be little or no difference 

 between the use of unity for the slope coefficient 

 or use of the estimated value of 0.86. The simi- 

 larity in the two estimates is apparent when 

 the actual fish lengths are substituted into the 

 two equations and the two sets of slopes are 

 compared (Table 2, columns 1 and 2). 



We fitted the Case I model to four additional 

 species to determine the extent they differed in 

 the length coefficient for the slope in the speed- 

 tail beat frequency relationship. Used in this 

 comparison were data we collected on Scomber, 

 and data presented in scatter plots of velocity 

 and frequency for individual Carassius, Salmo. 

 and Le?tc/so!« by Bainbridge (1958). We used 

 the XY digitizer to transcribe Bainbridge's data 

 onto cards. We may have failed to interpret 

 correctly some of the overlapped points in his 

 graphs but the effect of these errors on the sta- 

 tistical parameters we estimated would be ne- 

 gligible. We did not use the data presented by 

 Magnuson and Prescott (1966), Yuen (1966), 

 or Fierstine and Walters (1968) because in these 

 studies the absolute speeds and the lengths of 

 the fish were unknown. 



Our estimates of the slope coefficient for the 

 speed-tail beat relationship, 5.,, varied from 0.76 

 in Salmo to 1.22 in Carassius, and the 90% 



Table 2. — Slopes for the speed-frequency relationship for individual Trachurus 

 when the general relationship is slope = 1.28 L^^^ (Case I) and when slope =: 

 0.86L (Case II) ; estimated minimum speed (V„) for each fish when V^ = 0.80L2/3; 

 lowest observed test speed (Fobs); the minimum tail beat frequency (F„) estimated 

 by substitution of Fq the Case II equation for each fish; and the lowest observed 

 tail beat frequency (fobs)- 



Length 

 (cm) 



= 1.28/." 

 Case I 



i = 0.86i 

 Case II 



= 3.98i 

 Case II 



258 



