VOL. LXXIxJ PHILOSOPHICAL TRANSACTIONS. 579 



the given equation for ap, p/), and piriy substitute z ± ^, '-—, andy + -^; 



and there will result the equation required, expressing the relation between z and 

 X the abscissae, and their correspondent ordinates y. 



3. Fig. 13. To change the ordinates, the abscissae remaining the same, draw 

 p'm an ordinate transformed, p'h parallel to the first abscissa ap, and meeting a 

 2d abscissa, of which pm is an ordinate in h: for the sines of the angles p'hp, hpp\ 

 and hp'p; p'pm, pmp\ and pp*m, write r, s, and t, r, s', and t'\ and for ap', 

 p'/)', and p'm, respectively z, x, and y ; then substitute in the given equation for 



AP, p/>, and pm, their respective values z (ap') ± — - X y, oc (p'p ± -j y, and 



-,y ; and there results an equation to the solid expressing the relation between 

 the 2 abscissae ap' and v'p' and the transformed ordinates p'm. 



From these cases, which are easily reducible to one, may be transformed any 

 given abscissae and their correspondent ordinates into any other containing given 

 angles, &c. with the before-mentioned abscissae and ordinates. In the properties 

 of curve lines, first published in J 7^2, is given a method of deducing the equa- 

 tion to any section of the solid,^ and in particular the case of deducing the equa- 

 tion to the projection of any curve on a given plane. From the principles given 

 in this, and the paper on centripetal forces, which the r. s. did me the honour to 

 print, can be deduced the fluxional equations, whose fluents express the relations 

 between the abscissae and their correspondent ordinates, of the curves described 

 by bodies of which the particles act on each other with forces varying according 

 to given functions of their distances. 



XIX. Experiments on the Congelation of Quicksilver in England. fVith further 

 Experiments on the Production of j^rtificial Cold. By Mr. Richard Walker, 

 P- 199- 



Exper. 1. — On December 18, 1788, a favourable opportunity offered of 

 beginning some experiments on the congelation of mercury. For this purpose 

 Mr. W. prepared a mixture of diluted vitriolic acid (reduced by water till its spe- 

 cific gravity was to that of water as I.5596 to l) and strong fuming nitrous acid, 

 of each equal parts. The glass tube of a mercurial thermometer, with its bulb 

 half filled with mercury, was provided, as a convenient method of ascertaining 

 when the mercury was congealed ; for if, after being subjected to the cold of a 

 frigorific mixture, the thermometer glass should be taken out and inverted, and 

 the mercury found to remain completely suspended in that half of the bulb now 

 uppermost, no doubt can remain of the success of the experiment; an hydrometer, 

 with its lower bulb half an inch in diameter, and ^ full of mercury, was also pro- 

 vided, in case any accident should happen to the other. 



4e2 



