1904.] 



A Radial Area-Scale. 



295 



shows that they form a large variety of spirals. An important point 

 to notice is that if, as usual, <f> denotes the inclination of a radius to 

 the tangent at its extremity, we have tan <£, i.e., rcW/dr, equal to 



which, in general, cannot be zero unless at the origin, when the curve 

 passes through the origin. This means that error is likely to be 

 perceptible when the curve quadrated is such that any tangent to it 

 passes through the point of radiation ; e.g., when the outside lines are 

 tangents to the curve ; or when the curve is re-entrant in such a way 

 that any one of the radiating lines gives three or four readings. The 

 design may be expected to give the best results, therefore, when 

 arranged so that the outside lines past through cusps or angular points 

 of the curve, and so that no radiating line crosses it more than twice. 

 For curves in which there are no sharp points, it is best, therefore, to 

 divide into two or three areas. 



There is no reason why the presence of points of contrary flexure 

 should be supposed to vitiate the results. For the equation for 6 

 giving the positions at which such points occur in the stated family of 

 curves, will be found to be, in general, of the sixth degree. 



Calculable deviations "from strict accuracy may be expected in the 

 case of nearly all the well-known regular curves quadrated by means 

 of the area-scale. If, as in the present pattern, the angle between the 

 outside lines is half a radian, these deviations will be found to be 

 insignificant, except in the cases of oval curves touching the outside 

 lines and not treated as suggested on p. 292. 



b + 2c8 + 3d& 2 



Y 2 



