VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 185 



curv.e, being A, its axis ab perpendicular to the horizon, and its ordinate bd 

 parallel to the same. There is to be found the relation between b^' or Bi and 

 di; supposing the point b to be infinitely near to B, aqd bd parallel to bd, also 

 vS parallel to ab. 



It appears from mechanics, that three powers are in equilibrlo, when they 

 have the same ratio as three intersecting right lines that are parallel to their 

 directions, or which are incHned to them in a given angle, being terminated by 

 their mutual concourse. Therefore, if nd denote the absolute gravity of the 

 particle ad, as it must be in a chain of uniform thickness ; then dS will repre- 

 sent that part of the gravity that acts perpendicularly on od, by which it 

 happens, (because of the flexibility of the chain moving about d,) that do 

 endeavours to reduce itself to a vertical situatioTi. Therefore if Sd, or the 

 fluxion of the ordinate bd, be supposed constant, the action of gravity exerted 

 perpendicularly on the corresponding parts of the chain do, will also be con- 

 stant, or every where the same. 



Let this be expounded by the right Ifne a. Again, by mechanics before 

 cited, vS the fluxion of the axis ab, virill denote the force, which is exerted 

 according to the direction of do, which is equivalent to the aforesaid endeavour 

 of the heavy line do to reduce itself into a vertical situation, and which prevents 

 its doing so. 



Now this force arises from the heavy line da drawing according to the direc- 

 tion do, and therefore caeteris paribus is proportional to the line da. Therefore 

 Sd, the fluxion of the ordinate, is to So, the fluxion of the absciss, as the 

 constant right line a is to the curve da. a. e. f. 



Corol. — If the right line td touch the catenaria, and meet the axis ea pro- 

 duced in t, it will be bd : bt :: (dS : So ::) a : curve da. 



Prop. II. Theor. — If to the perpendicular ab as an axis, with vertex A, an 

 equilateral hyperbola ah be described, whose bemiaxis ac is equal to a ; and to 

 the same axis and vertex, a parabola ap be drawn, whose parameter is equal to 

 four times the axis of the hyperbola ; and if the ordinate hb of the hyperbola 

 be continually produced, till hf be equal to the curve ap ; I say the curve fad, in 

 which the points f and d are found, (supposing bd = bf) will be the catenaria. 



Make ab := jr ; then b^ := i, and bh = \^'2ax -\- xx. Whence, by the me- 

 thod of fluxions, the fluxion of bh = —. ^ — = in h. Again, because the 



parameter of the parabola ap is = 8a, it is bp = /8«x. Whence vp, the 

 fluxion of BP, will be ^^. So that the fluxion of the curve ap=p/3— 



VnpXnp-\--enX en = \/ '^^ + .;- = ^ - ''''' "*" '^ ' " , which by multiplying 



VOL. IV. B B 



