Meditation on the Idea of Poncelet's Theorem. 307 



five miles from the outlet of Ogiven Lake, Carnarvonshire, 

 North Wales. 



From Snowdon, 100 yards above the Ceunant Maur Fall, Car- 

 narvonshire, North Wales. 



From the junction of the River Kinder and Sett, Derbyshire. 



From the Brent, two miles below Apperton Bridge, 



From the Thames, a quarter of a mile above Teddington lock. 



From the Mardyke, near Stifford, Essex; falls into the River 

 Thames at Grays. 



From the River Lea, Essex, 300 yards above the East London 

 Water Works. 



From brook emptying itself into the Wrike, Melton Mowbray, 

 Leicestershire. 



From Ladies Well Spring, Pitlour, Fifeshire, Scotland. 



From Garden Brook, Pitlour, Fifeshire, Scotland. 



Although in my experiments I have not yet met with a sand 

 from the source or bed of a stream or river that does not contain 

 arsenic, and perhaps antimony also, yet, judging from the ex- 

 ceeding variability in the quantity in each, I am far from think- 

 ing that sands will not be found, though I should say they will 

 be few in number, that will not contain these metals. 



XXXIX. Meditation on the Idea of Poncelet's Theorem. By 

 J. J. Sylvester, M.A., F.R.S., Professor of Mathematics in 

 the Royal Military Academy, Woolwich*. 



HITHERTO Poncelefs theorem has been regarded as a 

 method sui generis and complete in itself; but in truth it 

 is' but the first germ or rudiment of a vast and prolific alge- 

 braical theory ; and not only so, but the principle which it con- 

 tains admits of applications of the utmost valine in various 

 dynamical and analytical questions, which it is surprising should 

 have been allowed to lay so long dormant. For the present, 

 however, I mean to confine myself to a very brief indication of 

 one direction in which the theorem admits of being generalized. 

 And first I will make a remark upon so simple a matter as the 

 extraction of the square root, which seems to have escaped obser- 

 vation, and at all events is so far from being generally known, 

 that two of the highest authorities for mathematical erudition in 

 this country whom I have consulted on the subject provisionally 

 accept it as new. 



Let r be an approximate value of */N ; then by that mode 

 of application of Newton's method of approximation to the equa- 

 tion a? 2 = N which is equivalent to the use of confinued fractions, 



* Communicated by the Author. 



