Astronomical and Nautical Collections. 355 



stired by m + n A", we should have = d^, and t = 



— / arc tangf / — A), consequently aJ {mn) < = — arc 



V^ 9/171 \ "V 7/t 



tang I /— A), and tang J (mn)t, = - ^ /— A and A = 

 \ V m V m 



— /— tang (^ (TO?i) + c ; and, for the correction of the 

 ■V n 



fluent, a~ + c, and A = a — /^ tang. (^ (»iw)0- 



V n 



There appears to be an oversight in a remark inserted among 



the Elementary Illustrations of the Celestial Mechanics, p. 145; 



where it is observed that " the whole time of the oscillation can 



never be sensibly affected by any small resistance proportional 



to the velocity ;" for, in fact, the coefficient y, in the expression 



of Laplace, being equal to V (^ ^^Y is in some degree 



affected by m, which expresses the resistance ; and the time is 

 affected by y, though Laplace has not investigated the precise 

 effects of a given resistance. That which is here inferred from 

 Captain Kater's experiments, liowever, would scarcely produce 

 a retardation of one fiftieth of a second in a year : and must, 

 therefore, be wholly neglected. 



If we are anxious to reconcile the existence of a retardation 

 proportional to the velocity, with the common theory of the im- 

 pulse of fluids, it will not be difficult to understand how the 

 one may possibly be derived from the other. We have only to 

 suppose the pendulum subjected to the influence of a very slow 

 current of air, in order to deduce a resistance nearly proportional 

 to the velocity v from another, which depends on (c ip v)~. For 

 it will appear, by considering the directions of the forces con- 

 cerned, that at the extremities of the vibration, while the velo- 

 city of the current exceeds that of the pendulum, and c — v 

 remains positive, the quantity 2c?; will denote a retarding force 

 throughout the motion, and that the portions c" and v^ will be 

 retarding in one direction and accelerating in the other, and 

 will have no sensible effect on the extent of the vibrations ; 



