VOL. XIX. 3 PHILOSOPHICAL TRANSACTIONS, lg3 



extremes. And therefore ck + ar and or — ar are the numbers hl or gp, 

 belonging to the given logarithm ch or cp. 



Carol. 3. — From the demonstration it is evident, that as hd the half sum of 

 the ordinates hl, pg, of the logarithmic curve, applied perpendicularly to ch 

 in H, is the ordinate of the catenaria ; so the half difference of the same hl, 

 PG, applied perpendicularly to ca in b, is the ordinate of the equilateral hyper- 

 bola, described with centre c and vertex a ; and therefore by Cor. 1, Prop. 2, of 

 this, is equal to the catenaria ad ; ?or y=is/ 2aa:-\-xx. And since it is shown 

 in the foregoing corollary, that ar also is the half difference of the right lines 

 hl, pg, it is plain that ar is equal to the portion of the catenaria ad. Whence, 

 by the way, a method is discovered, from the chain ad being given, to find c the 

 centre of the conterminate hyperbola, or that point in the asymptote of the 

 logarithmic curve gl. For if ar be taken equal to the chain ad, and from the 

 middle point of the right line br, a perpendicular to it be raised, this will meet 

 BA the axis of the chain in the point required c, as very plainly appears. For 

 thus CR will be equal to cb. 



Carol. 4. — Hence also it follows, that if the angle bdt be made equal to acr, 

 the right line dt will touch the catenaria in d. For thus, in the similar tri- 

 angles DBT and CAR, it will be db : bt :: ca : ar, or the curve ad, which is 

 equal to it. And therefore by Corol. Prop. 1 of this, dt touches the catenaria. 



Caral. 5. — It also follows, that the space achd is equal to the rectangle of 

 CA and AR. For because ayd, by Prop. 4, is equal to the triangle under CA and 

 AD — BD = (by Corol. 3 of this Prop.) ar — ay =: yr, the proposition is plain. 

 And because ca is given, it is evident that the space achd is as the curve ad, 

 or its fluxion ad is as the fluxion of this Dd. 



Caral. 6. — If through the point k, where cfi meets hd, a line kz be drawn 

 parallel to ph, meeting the right line ac in z, and ce be taken equal to half the 

 sum of BC, cz ; the point e will be the centre of equilibrium of the curve fad. 

 Upon FAD let it be conceived, that the upright superficies of a cylinder is 

 erected, and cut by a plane through ph, at half a right angle with the plane of 

 the curve fad. This superficies will expound the moment of the curve fad, 

 when librated on the axis ph ; and its fluxion is dh X d</ + pp X f/= 2bc 



a.i + xx 2aax + iax.r+2xx.v 



XAD = 2Xa-\-xX : ' = ; • — ^ = . + 



v2ax + xx V2ax + xx V2ax + xx 



aax + aix 3axx+2xxx 



~ /- , + /■' , of which the fluent is a X b d + a v'2 a x -\- xx + 



V2ax + xi V2ax + xx '^ i i i 



*' '^ lax -\- xx=: caXbd-I-cb X ad. Wherefore ca X bd + cb x ad is 

 equal to the aforesaid cylindrical superficies, (for they are nascent together,) 

 which is equal to the moment of the curve pad, when poised on the axis ph. 

 VOL. IV. C c 



