386 PHILOSOPHICAL TRANSACTIONS. [aNNO I788. 



Exam. 2. Let s be the centre of the logarithmic spiral, then will the equa- 

 tion be a X sp = a X D = SY = p, and consequently the resulting equation 

 fl X D = - X p3 whence — X d = /> an equation to a logarithmic spiral having 

 the same centre. 



Exam. 3. Let t and c be the semi-conjugate axes of a conic section, and s its 

 centre; then will the equation expressing the relation between the distance d and 



perpendicular p be d^ ± — j- = t*^ ± c^; forp write as before —, and there re- 



suits the equation d^ ± — 5-7 = t'^ + c^, an equation to a conic section of the 



same name, of which the transverse and conjugate diameters are respectively two 



roots {x) of the equation x'^ ± — —- = t^ + c*^, because in this case /> = d. 



The sum or difference of the squares of the transverse and conjugate diameters, 

 in all the resulting equations, will be the same. 



Corol. In every equal distance, the chord of curvature passing through the 

 centre of force is the same; for the forces in that direction, and the velocities at 

 every equal altitude are the same. 



Prop. 3. — 1. Fig. 6 and 5. Given an equation a = O, expressing the relation 

 between the absciss sm = a: and ordinate mp = ^ ; to find the equation expressing 

 the relation between sv = V {x'^ -f y^) and sy = p, the perpendicular from s on 

 the tangent py. From the equation a = find x = By, which substitute for x 

 in the equation (d^ -{- y'^)^ X v = xy ± xy deduced from the similar triangles p/o, 

 MTP, and STY, where to = x and po = jr; let the resulting equation be c = 0; 

 reduce the three equations a = O, c = O, and x^ -\- y'^ = sp^ = d^ into one, so 

 that the unknown quantities x and y may be exterminated, and there results an 

 equation expressing the relation between d and p. 



Corol. Hence, from the equation expressing the relation between x and y, 

 the absciss and ordinate of a curve, can be deduced an equation expressing the 

 relation between the distance sp and perpendicular sy ; and from the equation 

 expressing the relation between the distance sp and sy can be deduced an equa- 

 tion expressing the relation between the distance sp and perpendicular sy from 

 the point s to the tangent py of a curve, whose force and velocity at every equal 

 distance is the same as in the given curve, but the direction different. 



2. Given an equation k = O, expressing the relation between sp = d and 

 sy = p ; to find an equation expressing the relation between sm =: a: and pm = y, 

 the absciss and ordinate of the same curve. In the given equation k = O for d 

 and p write respectively \^ {x"^ + z/^) and - ff^.~^,. , and there results a fluxional 

 equation l = of the first order, of which the fluent expresses the general rela- 

 tion between x and y. 



Corol. If in the given equation for p be written wp', there results the equa- 



