On the Problem of the In-and-circumscribed Polygon. 337 



the battery used to excite the machine would ignite nearly three 

 times the length of the same wire. Now I venture to throw out 

 for the consideration of those engaged in the interesting and 

 important investigation of the correlation and conservation of 

 physical forces, and who have more time to prosecute their in- 

 quiries than I can possibly command in the occasional intervals 

 snatched from active business occupations, that it would be very 

 curious to ascertain the relation between the thermal characters 

 of the current employed and the current produced ; and if, by 

 any augmentation in the size of the apparatus, the secondary 

 thermal effects should be ever made to equal those of the pri- 

 mary current, what would be the intermediate relationship of the 

 magnetism which would then almost appear to be an extraneous 

 development. I do not throw out this hint by way of express- 

 ing any want of confidence in a theory, the truth of which, if 

 not already established, is pretty nearly so, but merely to sug- 

 gest what appear to me new conditions and relationships which 

 require to be harmonized with each other, and included in the 

 general consideration of the phsenomena. 



XLVI. 071 the Problem of the In-and-circum^mbed Polygon. 

 By the Rev. George Salmon, Trinity College, Dublin*. 



THE following is the conclusion of my investigations on this 

 subject published in the last Number. It is found by the 

 method there described, that if two sides of a triangle touch a 

 conic U, and the third side a conic «U + Z»V, if the two base 

 angles move on the conic V, then the locus of the vertex will be 

 one or other of two conies touching the four common tangents 

 of U, V, and whose equations are of the form 



where \ is given by the quadratic equation 



ff(«i-/3«)\2 + «(4Aa + 20i)X/^-ZiV=O. 

 where «=4AA', /3=02-4A0'. 



Now if it be required to find the locus of the free vertex of a 

 polygon, all whose sides tovich U, and all whose vertices but one 

 move on V, this is immediately reduced to the last question, 

 since the line joining the two vertices of the polygon adjacent to 

 that whose locus is sought, touches a conic whose equation is of 

 the form «U + iV = 0. 



The locus will therefore always be of the form 

 AA'\2V + VF + /i2u=o. 



* Communicated by the Author. 

 Phil. Mag. S. 4. Vol. 13. No. 87. May 1857. 2 A 



