BY MARTIN GARDINER, C.E. 67 



5. It is evident we can investigate in like manner (and that 

 the locus of will be a circle) when, instead of one closed n'gon> 

 we are given any number of closed n'gons, and that the sum of 

 the products of the rectares of their derived figures and given 

 numbers of known signs, is to be of a given magnitude. 



And it is also evident the locus of is a circle when the sum 

 of the products of the rectares of some of the derived figures and 

 given numbers, has a given ratio to the sum of the products 

 of given numbers and the rectares of the remaining derived 

 figures. 



6. Again (owing to the nature of the investigation, and to 

 our knowledge of the relative properties of approximate figures) 

 it is evident the principle of continuity justifies the extension of 

 our results to the more comprehensive propositions in which given 

 straight lines are replaced by curves of any kind whatever in a 

 plane. And (remembering that the curvatures at points in curves 

 are proportional to the angles between tangents at the extremities 

 of equally long elements} we may announce the following important 

 porisms : 



PORISM I. 



Given any lot of closed figures of prescribed formations in a plane, 

 then will the locus of a point o be a determinable circle when the 

 sum of the products of given numbers and the rectares of the derived 

 figures of the lot in respect to this point is of a given or determinate 

 magnitude %. And, for all values of %, the centre of the determin- 

 ate circle (whose circumference is the locus of o) is a fixed point, 

 with which the locus of o is coincident when 2 is a minimum, and 

 which point is the mean centre of the curvatures when the given 

 figures are closed curves and the given numbers all equal. 



PORISM II. 



Given a lot of closed figures of prescribed formations, and given 

 also a second lot of closed figures of prescribed formations ; then will 

 the locus of a point o be a determinable circle when the sum of the 

 products of given numbers and the rectares of the derived figures of 

 the first lot in respect to the point, has a given or determinable 



