44 PHILOSOPHICAL TRANSACTIONS. [aNNO J 735. 



numbers m and n, be described in the same plane, the greatest number of points 

 in which these lines can intersect each other, will be mn, or the product of 

 the numbers which express the dimensions of the lines, or the orders to which 

 they belong. 



II. In the next part, theorems are given for drawing tangents to all the curves 

 that were described in that treatise by the motions of angles on given lines. 

 Their asymptotes are also determined by more simple constructions than those 

 which are subjoined to their descriptions in that treatise. Of these we shall 

 give one instance here. 



Suppose the invariable angles, fig. 1 and 2, pi. 3, pcg, ksh, to revolve about 

 the fixed points or poles, c and s. Suppose the intersection of the two sides 

 OF, SK, to be carried over the curve bom, whose tangent at the point q is sup- 

 posed to be the right line ae ; and let it be required to draw a tangent at p to 

 the curve line described by p the intersection of the other two sides cg and sh. 



Construction. — Draw qt constituting the angle saT, equal to caA, on the 

 opposite side of s«, that aA is from cq; and let qt meet cs, produced if neces- 

 sary in t. Join pt, and constitute the angle cpn equal to spt, on the opposite 

 side of cp, that pt is from sp; then the right line pn shall be a tangent at p, 

 to the curve described by the motion of p, which is always supposed to be the 

 intersection of cg and sh. 



The asymptotes of the curve, described by p. are determined thus. Find, 

 as in the abovementioned treatise, when these sides become parallel, whose 

 intersection is supposed to trace the curve; which always happens when the angle 

 cas becomes equal to the supplement of the sum of the invariable angles fcg, 

 KSH, to four right ones; because the angle cps then vanishes. Suppose, in 

 fig. 3 and 4, that when this happens, the intersection of the sides of, sk is 

 found in a. 



Constitute the angle saT equal to caA, as before, and let aT meet cs in t. 

 Take cn equal to st, the opposite way from c that st lies from s. Through n 

 draw DN parallel to cg or sh, which are now parallel to each other; then dn 

 shall be an asymptote of the curve described by the motion of p. 



If instead of a curve line bqm, a fixed right line ae be substituted, then the 

 point p will describe a conic section, whose tangents and asymptotes are deter- 

 mined by these constructions. In this supplement, it is afterwards shown how 

 to draw the tangents and asymptotes of all the curves which are described in 

 the abovementioned treatise by more angles and lines. 



III. The same method is afterwards applied to draw tangents to lines described 

 by other motions than those which are considered in that treatise ; of which the 

 following is an instance. Suppose that the lines CP and sp, fig. 5, revolve 



