FISHERY BULLETIN; VOL. 69, NO. 2 



tional Marine Fisheries Service, La Jolla, Calif.) , 

 Area = 0.0281 lL'°'--» (lifting area of pectorals 

 of T. symmetricus was equal to twice the pec- 

 toral fin area), and Df — 1.03 (the density of 

 T. trarhurus from Alexander (1959b), a closely 

 related species to T. symmetricus), we obtain 

 an estimate of Fo = 0.86L°<'5 for Trachurus. 

 This value is considerably below lOL"^" obtained 

 for Euthynnus by Magnuson (1970) . The min- 

 imum swimming speed of Trwhnrus would be 

 expected to be lower than that of Euthynnus be- 

 cause Euthynmis lacks a swim bladder and has 

 a high specific gravity. Indeed, the minimum 

 speed of Euthynnus is close to the endurance 

 speed of many fishes with swim bladders (Mag- 

 nuson, 1970). 



In all other species except Carasshis estima- 

 tion of minimum speed was not possible because 

 we had few or no estimates of the variables 

 required in Magnuson's equation. In Carassius 

 we used the specific gi'avity for carp, 1.002 g/cc 

 (Alexander, 1959a), the pectoral fin area re- 

 lationship of Area = 0.02811Li"-^ from data we 

 collected on seven Carassius 4.6 to 22.5 cm total 

 length (the lifting area of the pectorals equaled 

 twice the pectoral fin area) and the length- 

 weight relationship of Ma = O.OOGSL^^" for the 

 seven Carassius. The estimate of minimum 

 swimming speed Vo for Carassius from these 

 data was 0.87L'"^^ This estimate was nearly 

 the same as the one estimated above for Tra- 

 churus and it had the same coefiicient of length. 

 Thus, in Trachurus and in Carassius the min- 

 imum speed coeflncient of length or, in our 

 equation, the intercept coefiicient 5,. was 0.65. 



That the length coefiicient for the intercept 

 was the same in Carassius and Trachurus sup- 

 ports the basic assumption of the Case II model, 

 that is, the existence of common length coefiicient 

 among different species. To further test this 

 assumption we estimated the intercept coeffi- 

 cient by fitting the combined data from all five 

 species listed in Table 3 to the reduced Case II 

 model 



<1.92. Although the limits were wide, the 

 estimate was very close to the other independent 

 estimate for Carassius and Trachurus and sug- 

 gests that the true value may be close to 2/3. 

 The evidence we presented supported the use 

 of the Case II equation and the use of 2/3 for 

 the length-dependent coefficient for the intercept 

 function and of unity for the length-dependent 

 coefiicient of the slope function. Thus, we fit 

 the reduced Case II model 



V = cc.U'^ + a,L * F 



to the data from each of the five species listed 

 in Table 3 and to that from two additional species 

 Triakis and Sardinops for which we had a small 

 number of ob.servations. The resultant equa- 

 tions were useful nonbiased predictive models 

 for the estimation of speed (V) from length 

 (L) and tail beat frequency (F) in each spe- 

 cies (Table 4). The regression lines for these 

 equations do not pass through the origin, how- 

 ever. They cut the abscissa before zero and 

 consequently the intercept terms are negative. 

 We pointed out previously that we believe the 

 existence of a negative intercept in the raw 

 data implied that the fish had a minimum swim- 

 ming speed below which they cannot swim by 

 beating only the caudal fin. Thus, to make the 

 model more biologically meaningful we adjusted 

 the elevation of the intercept function to cor- 

 respond to the theoretical minimum swimming 

 speed, Vo. (It should be remembered that the 

 length-dependent slope of Vo was about the same 

 as the length coefiicient, a.^, and it was this simi- 

 larity that led us to use 2 3 as the intercept 

 coefficient.) 



To express the Case II model in terms of min- 

 imum sjjeed we used Fo estimated from Mag- 

 nuson's equation to solve for a minimum tail 

 beat frequency, Fn, and expressed the final re- 

 lationship in the form 



Vo 



KF — Fo. 



a^L 



+ a,L* F 



The estimate a., for the combined data was 

 0.68 with 90% confidence limits of —0.56 Ka^ 



In species other than Trachurus and Carassius, 

 a theoretical estimate of Vo was not possible 

 and consequently we assumed Fo was propor- 



260 



