44 The Three Sections, Tangencies, 



NOTES. 



1, It is evident that the porisms in the Transactions for 

 1859 can, with many others, be derived from the porismatio 

 states of the data of this or the first Generating Problem of 

 the Three Sections. 



2, And very probably, the Greek geometers derived from 

 this and other kindred problems, by means of projections, &c., 

 part of their ' porismatic knowledge' which is now known as 

 the ' anharmonic properties of pencils and divisions.' 



3, In the investigations of the limits of the angles 6 right 

 in problems 1 and 2 of my paper in the Transactions for 1859, 

 it would be well to omit all the Avords fi-om ' And it is more- 

 over evident,^ &c., and substitute the following : — And when 

 the circles iBC circumscribe the portion of circle ACH which 

 is not within or outside both the circles BAi, the limiting 

 values include between them all values of 6 right, and no 

 others, for which CO and BO are imaginary; but when the 

 circles iBC circumscribe the portion of circle ACH which is 

 Avithin or outside both circles BiA, then the limiting values 

 have oiitside them aU values of 6 right, and no others, for 

 which CO and BO are imaginary. At the limits the lines 

 CO are real and coincident. 



THE TANGENCIES. 



FIRST SOLUTION. 



(See Plate.) 



To describe a circle to touch three given circles A, B, C. 



ANALYSIS. 



Let D, E, F, be the respective points of contact of the 

 required circle Avith the circles A, B, C. Then DE passes 

 through O a knoAvn centre of similitude of circles A, B ; and 

 DF passes through P, a known centre of similitude of the 

 circles A, C. 



Now if D' be any assumed point in circumference A, and 

 that E', F', are the dissimilar points in which OD' and PD' 

 cut circumferences B and C ; then PD'.PF' = PD.PF, and 

 OD'.OE' = OD.OE; and it folloAvs that the circles D'E'F', 

 DEF, have PO as radical axis. 



Let d' , e' , and/', be the other points in which the knoAvn 

 circle D'E'F' cuts the circles. A, B, C. It is evident the 



