52 THE ROYAL SOCIETY OF CANADA 



values from all the scales were uniform. This proved to be almost 

 immediately below a' point midway between zero and the point of 

 intersection of the curve and the base line, and it also proved to be 

 the same as the point at which the curves for the tail and pectoral 

 scales respectively met when produced. 



Calculations are made by means of the movable curve rotating 

 about the point of convergence in the following manner. A scale is 

 taken from a fish of length L, and the distance from the focus to the 

 limits of the successive zones measured, the final one being from the 

 focus to the anterior tip of the scale. These are denoted Xi, X2, 

 X3, X4 X. 



The curve is adjusted until it intersects X vertically above L. 

 The values Li, L2, etc., corresponding to Xi, X2, etc. can then be read 

 off (see figure 2). 



This method compensates for (1) diff'erential growth of the given 

 dimensions of the scale compared with the length of the fish, and (2) 

 differences in the time of origin of various scales according to size. 

 It does not compensate for differences between individual fishes in 

 the character of the differential growth of the scales and in the time 

 of origin of the scales. It is not apparent how a convenient method 

 of compensating for these can be devised. 



Other sources of error are as follows. There may be temporary 

 variation in the growth rate of the scales as compared with the fish. 

 The relative rate of growth may vary during the course of the yearly 

 cycle, as the varying distance of the circuli from each other shows that 

 the relative growth of the parts of the scale varies. Temporary con- 

 ditions of another character may change the relative rate of growth 

 of the scale and of the entire fish. If a scale be removed or dislocated 

 the new growth of scale in the scale pocket is for several years more 

 vigorous than in the case of neighboring scales, as is shown by the 

 greater distinctness of the yearly zones. In this case the removal 

 or dislocation of the scale acts as a stimulus to growth and it is probable 

 that other local or temporary causes may act in a similar way. If scales 

 are taken from different regions and the results averaged the error due 

 to local causes will tend to be eliminated. Error from temporary 

 causes affecting all the scales cannot be corrected, so far as can be seen. 



The probable error from any or all of these sources must be very 

 small. One can confidently count on the error from all methodical 

 sources being less than half a centimetre in the length of the fish, 

 by the method that has been described. Since individual variation 

 in the length of fishes of the same age is so great that measurements 

 to the nearest centimetre are ordinarily sufficient for statistical pur- 

 poses, a possible error of half a centimetre is not of great importance. 



