50 THE ROYAL SOCIETY OF CANADA 



in the case of the curve figured, if it is found that during the second 

 year the scale grew from 0-7 mm. to 1-5 m.m. giving a width of 0-8 

 m.m. for the zone, this would correspond with a growth in length of 

 the fish from 7-5 cm. to 13-5 cm. The great disadvantage of this 

 method is the difficulty of obtaining from the fish a scale of the proper 

 length showing the zones clearly. It is therefore important to have a 

 method that can be used for almost any scale, and for that we must 

 determine the regularity in the variation of the curves for scales of 

 different sizes. 



Curves representing the relation of very large scales and of very 

 small scales to the length of the fish would be similar in character to 

 the one shown, but would be placed differently, those for the large 

 scales above, and those for the small scales below, the curve shown. 

 We have in fact determined the curve for certain small scales, called 

 pectoral scales, situated near the pectoral fin. The data for this 

 determination are summarized in the lower series of dots in the 

 figure (1), and the curve itself, which is shown passing through these 

 dots, proves to be of essentially the same character as that for the 

 tail scales. 



The divergence of such curves from each other increases with 

 increase in the size of the fish, that is, there is less divergence near the 

 points on the base line where the curves commence. They all, in 

 fact, appear to converge at a point below the base line, at least all 

 in the same series, for there are indications that the series of scales 

 along the lateral line from the tail to the pectoral fin, which is the 

 one we are considering, differs in this respect from other series, as 

 for example, those farther from the lateral line. A scale from the 

 dorsal part of the tail may appear considerably later than the pectoral 

 scales and nevertheless surpass them in growth. 



The point of convergence of the curves in our series may be found 

 by determining at least two of the curves and projecting them until 

 they meet. 



An alternative indirect method has given the same result. The 

 curve, which was determined for the average tail scale, was made 

 movable by cutting it from a piece of thin wood or cardboard. When 

 this movable curve was made to rotate about the point where the 

 curve intersects the base line and when length calculations were made 

 with scales of different sizes, it was found that the values for the 

 first year's growth of a fish were much lower for large scales than for 

 small ones. When the curve was made to rotate about the point 

 where it intersects the vertical line through the zero point on being 

 produced, the values were higher for the large scales than for the small 

 ones. By trial a point was found as a centre of rotation, for which the 



