184 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



The application of these improvements will be easily made by the intelligent 

 reader, who is acquainted with what has been before written on the subject. 

 But there is a theorem, demonstrable by what is proved in art. 8, so remark- 

 able, that he cannot conclude this disquisition without taking notice of it. 



] 1. Let Ipqn, fig. 9, be a circle perpendicular to the horizon, whose highest 

 point is /, lowest w, and centre m; let p and q be any points in the semicircum- 

 ference Ipqn; draw ps, ^^ parallel to the horizon, intersecting Imn in j and /; 

 and, having joined Ip, pt, make the angle Ipv equal to Itp, and draw rv parallel 

 to qt, intersecting the circle in r, and the diameter Imn in v. Let a pendulum, 

 or other heavy body, descend by its gravity from p along the arc pqrn: the body 

 so descending will pass over the arc pq exactly in the same time as it will pass 

 pver the arc rn ; and therefore, qt and rv coinciding when It is equal to //», it is 

 evident that the time of descent from p io q will then be precisely equal to half 

 the time of descent from p to nl 



And it is further observable, that, if pqn be a quadrant, the whole time of 



descent will be = y^ X (ie + 1 ^ e'^ — 2c); the radius hn, or mn, being = 

 a; and b being put for (l6-riy feet) the space a heavy body descending freely from 

 rest falls through in one second of time. 



In general ns being denoted by d, and the distance of the body from the line 

 ps, in its descent, by x, the fluxion of the time of descent will be expressed by 



Vliad - d' -I2I'- 2d). x-x-r the fluent of which, corresponding to any value 

 of X, may be obtained by art. 7. By which article it appears, that the whole 

 time of descent from any point p will be = -7==—^ r X (e + 2ae — pn — bs). 



Vbd X (2o — a) ^ ^ r J 



The semi-transverse ac (fig: 7) being = ns; 

 the semi-transverse eg (fig. 8) = np; 



and the semi-conjugate in each figure = ps. 

 After writing the above, Mr. L. discovered a general theorem for the rectifi- 

 cation of the hyperbola, by means of two ellipsis; the investigation of which he 

 purposed to make the subject of another paper. 



XXX F II. On the Management of Carp in Polish, Prtissia. By J. Reinhold 



Forster, F.A.S. p. 310. 



Though the carp is now commonly found in ponds and rivers, and generally 

 thought to be a fresh water fish,* the ancient zoologists ranged the same among 



• I have great reason to think, that many other fish, which, it is commonly conceived, can only 

 live in the sea, may also exist, at least for several years, and perhaps breed, in fresh water. The 

 tmelt or sparling (salmo eperlanus Linnaei) never comes up our rivers, but for a short time ; and 

 then does not penetrate much farther than whera'the water continues to be brackish. I have how- 

 ever been informed by Sir Francis Barnard, the late governor of New England, that in a large pool 



