of Curves by their Curvature. 11 



convex above the plane surface level. At any depth d the 



curvature c must equal — , where A is the density of the 



v 



liquid (or difference of density of the two liquids if one forms 

 a drop in the other) , and t the surface-tension measured in 

 gravitation units. Thus if A means grains per cubic inch, 

 t is grains per linear inch. If a bubble is formed on a ring 

 or between two rings, the effect of gravity is so minute that 

 the curvature is practically the same at all levels, and the 

 curves that are formed are the well-known roulettes of the 

 foci of the conic sections, whose equations are known. 



In order to avoid the loss of time which results from the 

 perpetual finding of reciprocals in order to determine the new 

 radius of curvature of each small step in the curve, I have 

 divided a rule so that the distances of the divisions from the 

 beginning of the scale are the reciprocals of the numbers 

 attached to them. Such a rule is adapted to measure the 

 smallness of a thing just as an ordinary rule measures its 

 bigness or size. Thus a large thing since it has very little 

 smallness is found by the rule to be measured by a very 

 small number, while a very little thing having a considerable 

 smallness is measured in the same way by a very large number. 

 The curvature of a line is measured by the reciprocal or the 

 smallness of its radius, and so the curvature of a line is read 

 off immediately by the rule. The curvature of a surface is 

 measured by the smallness of two lines which start at the 

 same point and in the same or in opposite directions, and so 

 the curvature of a surface of revolution is found by inspection 

 of the two curvatures upon the rule. In order to be equally 

 ready to deal with the case when the lines are measured in 

 opposite directions, and with a further object not yet apparent, 

 I place the zero of the scale, marked oo , somewhere about the 

 middle and divide it each way. Then, if the two radii are 

 found on the same side of the co or centre, their values are to 

 be added, and if on the opposite side one is to be subtracted 

 from the other in order immediately to find the total curva- 

 ture. The only gain apparent at present is the great saving 

 of time resulting from an obvious construction. The increase 

 in accuracy depends upon the abolition of all accumulated 

 errors of compass setting, which are of three kinds : — (1) The 

 compass may not be set to the exact radius required ; (2) the 

 pencil- or pen-point may not be placed exactly upon the end of 

 the line just drawn, producing a step ; and (3) the needle- 

 point may not be set exactly upon the former radius-line, pro- 

 ducing an angle invisible no doubt, but still existent in the 

 curve. The first of these applies to the method of this paper, 



