OF LOCAL THEOREMS AND PORISMS. 339 



of our results, and we frequently arrive at loci belonging to the 

 conic sections. Sometimes, however, the simplification is still 

 greater, and we are surprised by recognising, amongst exten- 

 sive classes of curves, possessed of some peculiar property, 

 some well known locus of the straight line or circle. 



The inquiries which follow are given nearly in the order in 

 which they occurred ; a different arrangement might have ap- 

 peared more systematic, but it would have had the great dis- 

 advantage of concealing the means by which the results were 

 arrived at. Several of the more restricted porisms and local 

 theorems, might have admitted of a geometrical dress ; but this 

 would have been inconsistent with the object I had proposed 

 to myself in the present paper. The greater part are, I be- 

 lieve, beyond the powers of geometry ; and this opinion, if al- 

 lowed to be correct, will perhaps, by the admirers of the an- 

 cient geometry, be admitted as some excuse for the present 

 attempt to add to its stores, by means so very foreign. 



There exist numerous classes of curves possessed of the fol- 

 lowing property : 



If we take any abscissa AE, Plate XXII. Fig, 1, and ordinate 

 BE, and if we make AF •=. BE, and find the new ordinate CF, 

 and repeat the same process n times, the nth ordinate HI shall 

 equal the first abscissa AE. It is easy to perceive, that if?/ = 4/x 

 represent ithe. equation of the curve, thep the equation deter- 

 mining the form pf -4/ will be 

 •^^ae Z-.X 

 As the classes pfcurves here alluded to, will frequently occur in 

 the following inquiries, I shall venture to bestow on them the 

 name of Periodic Curves, which was suggested by the similar 

 name assigned by Mr Hersckel to the function which satis- 

 fies the equation just given. It wjill also be convenient to ap- 



