1S6 PHILOSOPHICAL TRANSACTIONS. [aNNO I697. 



^ax-\-xx 



both numerator and denominator into ^2a + x, becomes And since 



HE is every where equal to ap, the fluxion of the right line hf, that is mh-\-sf, 

 will be equal to -^ "^ ^^- . But it is already found that 7nh = -g f +-^-^ , ^ 



V'2ax + xx •' V'Jaj + jrx 



Whence 5 /"= — =- — ==, which is the fluxion of b f the ordinate to 



V '2 a X + X X 



the axis of the catenaria. Therefore the fluxion of the curve a f, or f/ 



/ — -pi — ; rr A / "■ X- 1-1 ax-\-3:x - , 



— V sj- + vs' = \J „ + .V- = — - - — , of which the fluent is 



V ^ a X + X X j^2 a T + X X 



^ 2cLV-\-rx, as just now found. Therefore af=^^ 2ax-\-xx. And it appears 



that the fluxion of the ordinate bf. or ~ ^, is to ,r the fluxion of the 



V2ax-|-.ij:' 



absciss ab, as the given line a to the curve af ; which is the property of the 

 eatenaria found above. Therefore the points ot the catenaria are rightly deter 

 mined, by the foregoing construction, a e. d. 



Corn/. 1. — From the construction it appears, that bf, the ordinate of the 

 catenaria, is equal to the parabolic curve ap, taking away bh the correspondent 

 ordinate of the conterminate hyperbola ah. 



Carol. 2. — From the demonstration it appears, that the curve of the catenaria 

 ap is equal to bh the correspondent ordinate of the conterminate equilateral 

 hyperbola. For since the fluxions of these lines are equal, and the lines them- 

 selves are nascent at the same time, it is plain they must be always equal. 

 Whence the chain being given, ac or a will be given also, being equal to the 

 semiaxis of the equilateral hyperbola whose vertex is a, and ordinate equal to 

 the absciss ab of the chain ad. 



Carol. 3. — All catenaria are similar to one another ; since thev are generated 

 by a like construction of like figures similarly posited. Whence two right 

 lines alike inclined to the horizon, drawn through the vertices of the chains, 

 will cut off similar figures, and portions of the chains which are proportional 

 to the right lines so cutting them oft'. 



Coral. A. — If the chain qad is su>;pended at the points e and d, which are at 

 unequal heights, the part of the curve fad continues the same as if it had 

 been suspended at the points f and d, which are equally high ; because it is all 

 one whether the point f be fixed to the horizontal plane or not. 



Carol. 5. — If the force of the chain, drawing according to the direction da 

 be denoted by or/; let it be divided, as is commonly known, into the force dS 

 according to a horizontal direction, and a force So according to a vertical 

 direction. Therefore in the extremity of the chain, the force of approaching 

 directly to the axis, is to the force of pcrpendiculir descent in the same ; or 

 the part of the sustaining force acting according to the direction bd, is to a 



