VO?.. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 4ij 



and other properties of it. Mr. Maclanrin had observed in 1719> that by in- 

 creasing tlie number of poles and angles beyond two, the dimensions of the 

 locus of p, did not rise above those of the lines of the second order, while 

 the intersections moved on right lines; and therefore he did not think it of use 

 then to take more poles than two, since by taking more, the descriptions be- 

 came more complex, without any advantage. When the intersections are car- 

 ried over curve lines, the dimensions of the locus of p rise higher, but the 

 curves described have double, or multiple points, as well as when two poles only 

 are assumed; and therefore this speculation is more curious than useful. How- 

 ever, he subjoins some of the theorems that he found on this subject, con- 

 cerning the dimensions of the locus of p, and the drawing tangents to it. 



1. If, in fig. 6, you suppose q and r to be carried over curve lines, of the 

 dimensions m and n respectively; then the point p may describe a locus of Imn 

 dimensions. 



2. If, in fig. 8, you suppose L, a, r, m, n, to be carried over curve lines of 

 the dimensions wi, n, r, s, t, respectively; then the locus of p may arise to 

 Qimnrst dimensions, but no higher; and if instead of lines revolving about 

 the poles, you use invariable angles, the dimensions of the locus of p will rise 

 no higher. 



3. He then assumed 3 poles, c, d and s, (fig. 10) and supposed one of the 

 angles snl, to have its angular point n carried over the curve an, while the 

 leg NG passes always through s, as in the description in the treatise of the ge- 

 neral description of curve lines, while the angles odr, rcp, revolve about the 

 poles D and c : he supposes also the intersections a and r to be carried over the 

 curve lines sa, gr, and that the dimensions of the curve lines an, Ba, gr. are 

 m, n, r, respectively; and finds that the locus of p may be of Smnr dimensions; 

 but that the point c is such, that the curve passes through it as often as there 

 are units in 2mnr. 



4. If any number of poles are assumed, so as to have angles revolving 

 about them, as about c and d in the last article, and the intersections are car- 

 ried over other curves, the dimensions of the locus of p will be equal to the 

 triple product of the number of dimensions of all the curves employed in the 

 description. 



5. If the invariable angles pnr, pmg, (fig. 11 ) move so, that while the 

 sides PN, PM, pass always through the poles c and s, the angular points n and 

 M describe the curves an and bm; and at the same time, the invariable angle 

 RDQ, revolve about the third pole d, so that the intersections r and a describe 

 the curves er and go. ; then the dimensions of the locus of p, when highest, 

 shall be equal to the quadruple product of the numbers that express the dimen- 



