48 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735. 



sions of the given curves an, er, oa and bm, multiplied continually into each 

 other. If more poles are assumed, about which angles be supposed to move, 

 as RDQ moves about d in this description, and the intersections of the sides be 

 still carried over curves, as in this example ; the dimensions, of the locus of p, 

 when highest, shall still be found equal to the quadruple product of all 

 the numbers that express the dimensions of the curves employed in this 

 description. 



6. Suppose that the three invariable angles pqk, klr, rnp, (fig. 12) move 

 over the curves gq, el, an, so that the sides pq, kl, pn produced, pass al- 

 ways through the poles c, d, s, and that the intersections of their sides k and 

 R, at the same time move over the curves fk and br ; then the dimensions of 

 the locus of p, when highest, shall be equal to the product of the numbers 

 that express the dimensions of the given curves multiplied by 6. If more poles, 

 with the necessary angles and curves, are assumed between c and d, as here 

 D is assumed between c and s, and the motions be in other respects like to 

 what they are in this example; then in order to find the dimensions of the 

 locus p, when highest, raise the number 2 to a power whose index is less than 

 the number of poles by a unit; add 2 to this power, and multiply the sum by 

 the product of the numbers that express the dimensions of the curves employed 

 in the description ; then this last product shall show the dimensions of the 

 locus of p when highest. 



The author is able to continue these theorems much further : but it is not 

 worth while, especially since there is not any considerable advantage obtained 

 by increasing the number of poles, above the method delivered in the above- 

 mentioned treatise, of the description of curve lines. On the contrary, the 

 descriptions there given, by means of 2 poles, will produce a locus of higher 

 dimensions by the same number of curves and angles, than these that require 

 3 or more poles; and are therefore preferable, unless perhaps in some parti- 

 cular cases. 



7. However, he has also found how to draw tangents to the curves that 

 arise in all these descriptions : of which he gives one instance, where 3 right 

 lines are supposed to revolve about 3 poles, and 2 of their intersections are 

 supposed to be carried over given curve lines, and the third describes the locus 

 required. 



Let the right lines ca, sn, dn, (fig. 13) revolve about the poles c, s, d ; 

 where that which revolves about d, serves to guide the motion of the other 

 two ; its intersection with ca moving oyer the curve oa, while its intersection 

 with SN moves over the curve fn. Suppose that the right line ah touches 

 the curve ca in a, and that the right line Aa touches the curve fn in n. In 



