'S5i') Astronomical and Nautical Collections. 



while, on the other hand, if the velocity of the pendulum tO'- 

 wards the middle of the vibration exceeds that of the current, 

 the force 2cv will retard the motion in one direction, and acce- 

 lerate it in the other, leaving only the constant resistance c^, and 

 the variable quantity i;-, which is proportional to the square of 

 the velocity. We obtain, therefore, for the extremities of the 

 vibrations, a force proportional to the simple velocity, and for 

 the middle, a constant resistance, and another force varying 

 simply as the velocity, the joint effect of all which must be a 

 resistance nearly such as has been inferred from Captain Kater'a 

 experiments, if the current moved at the rate of about half an 

 inch in a second, which would have been scarcely perceptible 

 to the senses. 



The question, however, regards not so much the distribution 

 of the resistance through the different parts of a single vibra- 

 tion, as its comparative value for the mean velocities of the suc- 

 cessive vibrations. Now, if the velocity of the current always 

 exceeds that of the pendulum, the only effective resistance will 

 be proportional to the simple velocity ; and when it is smaller 

 than the greatest velocity of the pendulum, the resistance will 

 approach more and more to the ratio of the square of the velo- 

 cities increased by a constant quantity ; and supposing the 

 velocity of the current to remain small and nearly uniform, 

 while the arc of vibration considerably diminishes, the whole 

 resistance will at first be more nearly as the square of the arc, 

 and if the arc be sufficiently diminished, the resistance propor- 

 tional to the simple velocity will at last remain alone. Hence, 

 it is easy to understand the variation of the constant coeffi- 

 cients in the different series of Captain Kater's experiments. 

 12 April, 1823. 



ii. Extract frovi a Letter to Professor Schumacheh, relating 

 to Bessel's Refractions. 



1 do not quarrel with you for your confidence in Bessel : but 

 I think you have not sufficiently attended to the limitation 

 under which he himself originally published his Theory of Re- 

 ftaction, Fundam. p. 55i " In distantiis d vcrticc non supet* 



