578 PHILOSOPHICAL TRANSACTIONS. [aNNO 1789. 



Find its value from the 2 given equations, in terms of ar, 2/, or z, multiplied into 

 its respective fluxions ; then its fluent, properly corrected, will be the length 

 of the arc required. If the angles are not right, they may easily be reduced to 

 them. 



4. The attractions of these surfaces, curves, &c. on a given point p, may be 

 deduced from the preceding principles of finding the attractions of each of the 

 parts in the directions of the first abscissa, which passes through the point p, the 

 2d abscissa, and the ordinates, and then finding the integrals of these increments. 

 From the method which determines the attraction of a body, surface, &c. on a 

 given point, can be determined the attraction of a body, &c. on any number of 

 points, and consequently the attraction of one body, &c. on another, &c. It is 

 sometimes advantageous to transform the first absciss, that it may pass through 

 the point attracted ; and the abscissae and ordinates, that they may be at right 

 angles to each other, &c. 



Pkob. 3. — 1. Fig. 12. Given an equation expressing the relation between the 

 2 abscissae ap and vp of a solid, and their correspondent ordinates pm, or ap', v'p\ 

 and p'rn ; to transform the first abscissa into any other lA. 



Let the abscissa lA begin from a point l of the first abscissa ap, and meet an 

 ordinate pm in the point h ; draw hp, and let the sines of the angles p/>yw., ^hp, 

 and pvh ; lpA, pAl, and plA, be denoted respectively by r, s, and t, and /, /, 

 and t' ; through a point h of the line vh draw p'h'm' parallel to pm, and make 

 lA = z, hh' = X, and h'm = y : in the given equation for ap, pp, and pm, sub- 



stitute respectively their correspondent values — + al (a), —7 + — (for pA = 



•-^ and pA' = pA + AA' = -7 ± x), and y ± —pr ± -p', then there results an 

 equation to the same solid, expressing the relation between the 2 abscissae z = lA 

 and X, and their correspondent ordinates y, 



1» 2. If the absciss lA does not begin from l, a point in the first given absciss 

 AP, but from m a point given out of it, it may be reduced to the preceding case, 

 by drawing from m a line mn = c to the plane of the 1st and 2d abscissae parallel 

 to the ordinates pm ; and from n to the 1st abscissa a line no = ^ parallel to the 

 2d abscissae, and substituting in the equation expressing the relation between ap, 

 pp, and pm for ap, vp, and pm respectively z ± ao {a), x ± b and y ± c ; and 

 there results the equation required expressing the relation between the 2 abscissae 

 z atid X, and their correspondent ordinates y, of which the 1st abscissa z passes 

 through the point m. 



2. To change the 2d abscissa pp into any other lA, the 1st abscissa and ordi- 

 nates remaining the same. In the preceding figure let l be considered as a 

 moveable point of the first absciss al, and the sines of the respective angles de- 

 noted by the same letters as before, and let lA = x, al = z, and hm =: y; in 



