CIRCLES AND OP SPHERES. 9 



exhibited in the two figures of Plate V. They are numbered as the points of con- 

 tact are numbered in circle A in both figures. Nos. 1 and 8 are convex to both 

 circles ; Nos. 2 and 7, concave to both ; 3 and 6, convex to circle A and concave 

 to circle F; 4 and 5, concave to A and convex to F. 



Problem 10. To draw a circle tangent to three given circles. — In Plate VI, let A, 

 B, and E, be the centres of the given circles. Draw the auxiliary circle B S, 

 having the same centre as the largest circle, and a radius equal to the difference 

 between the radii of the largest and of the smallest of the given circles. Draw, 

 also, the auxiliary circle A G with a radius equal to the difference between the 

 radii of the given circle A F and of E D, the smallest of the given circles. Draw 

 through E, the centre of the smallest circle, by Problem 6, a circle, as ESQ, tan- 

 gent to both these auxiliary circles. The centre, O, of this will be the centre of 

 one of the required circles. For D E = S C = F Q. Therefore, O D = O C = OF. 



By a construction entirely analogous, all the other solutions of this problem will 

 be readily obtained. Sometimes, the radius of the auxiliary circle is longer than 

 the radius of one of the circles by the radius of the smallest circle. 



It will be found that there are eight solutions to the problem, and they are all 

 exhibited to the eye in Plate VII. In each case, as in Problem 9, the centre of 

 the required circle is found by obtaining the centre of an auxiliary circle whose 

 radius differs from the required one (either less or greater) by the radius of the 

 smallest of the given circles. 



The following statement exhibits the auxiliary circles from which the different 

 solutions are obtained. The numbers refer to the required circle, whose point of 

 contact is numbered in Plate VII. 



Prom the auxiliary circles A G and B S are obtained, by Prob. 6 . Nos. 1 and 2. 



Prom B S' and A G' Nos. 3 and 4. 



From A G and B S' Nos. 5 and 6. 



From B S and AG' Nos. T and 8. 



It will be found on examination that the eight solutions are made up by tangent 

 circles, which are situated as follows in reference to the three given circles, viz: — 



No. 1, convex to the three circles. 



No. 2, enclosing or concave to all. 



Nos. 3, 5, and 7, concave to one and convex to the other two. 



Nos. 4, 6, and 8, convex to one and concave to the other two -circles. 



