68 GEOMETRICAL RESEARCHES, 



invariable ratio to the sum of the products of other given numbers 

 and the rectares of the derived figures of the second lot in respect to 

 this same point. 



7. These porisms give immediate intimation of numerous 

 interesting theorems, of which the two following are ex- 

 amples : 



THEOREM I. 



The rectare of the derived figure of any conic in respect to any 

 point in the circumference of a circle having the line joining the 

 foci as diameter, is equal the rectare of the circle. 



THEOREM II. 



If any number of conies have a common focus ; then will the 

 locus of a point o be a determinable circle, passing through this 

 focus, when the sum of the rectares of the derived figures of the 

 conies in respect to the point o is equal to the sum of the like 

 rectares of the circles having the transverse axes of the conies as 

 diameters. 



8. In respect to the general problem, it is evident that when 

 the given data is wholly or partly curve, the exact locus of 

 cannot be (unless in some particular cases) obtained without the 

 aid of the infinitesimal calculus. 



It is also obvious that in cases in which some points of the 

 given data are at infinity, the co-ordinate methods will afford the 

 best means of actual solution, though of course the principle of 

 continuity justifies us in predicting the nature of the locus, even 

 when the manner of approximating to its position (as indicated 

 in our general investigation) may not be intelligible to our 

 limited understanding. However, to clear all doubt on this 

 point, we can easily find the equation of the locus of O 

 without paying attention to the rectare of the given figure 

 A I A a ....A n A l . 



This may be done in various ways, but the following is 

 sufficient : 



