April i6, 1895.] 



SCIENCE. 



451 



only because the recent deatli of Tclieby- 

 chev. followed in less than two months by 

 that of Cayley, gives them now a special 

 pertinence, but because it is of interest to 

 compare one witli what is given on ' ti-am 

 motion ' in Kempe's ' How to Draw a 

 Straiglit Line,' and the other with its repro- 

 duction by no less a master than Clifford 

 on pages 149, 150 of his Dynamic, whence 

 I add figure 2. 



•' Robert's theorem of 3-bar motion takes 

 the following elegant form : Take a triangle 



Bl ~ CI ^ 



ABC and a point and through O draw- 

 lines parallel to the sides as in the figure, the 

 3 shaded A 's are of course similar to ABC. 

 Now imagine a linkage composed of the 

 shaded A's and the bars AAj, AA3, BB3, 

 BB,, CC,, CC„ pivoted together at A, B, C, 

 Aj, A3, B3, Bj, Co. Cj, O ; then, however, 

 the figure is moved [of course A3, B3 do 

 not continue in the line AB, etc.] , the tri- 

 angle ABC will remain similar to the shaded tri- 

 angles ; and if in any position of the figure we 

 fix the points A, B, C, then the point O will 

 be movable in a curve, viz.: we have the 

 same curve described by O considered as the 

 vertex of OA3 B3, where the two radii are 

 AA3, BB3 — by O considered as the vertex 

 of OA2 Cj, etc. — and by O considered as 

 the vertex of OB, C,, etc." 

 Cambridge, Feb. 22, 1876. 



" The porism is venj prettj' ; it was new 

 to me, though I think it ought not to have 

 been so. Look at the theorem thus: Im- 

 agine a plane, two points thereof, A, C 

 moving in fixed lines Ox, Oy. Describe the 

 circle OAC, which consider as a circle fixed 



in the plane and movable with it. Then 

 the theorem is that any point B of this 

 circle moves in a line OB through O. In 

 particular B may be the opposite extremity 

 of the diameter through A, and we have 



then the points A,B moving on the lines 

 Ox and OB at right angles to each other, 

 viz. : the general case of a plane moving two 

 points thereof on two fixed lines is reduced 

 to this well-known particular case. And 

 the theorem comes to this, that dividing 



the rod AB at pleasure into two parts AM, 

 MB, and drawing MC at right angles, and a 

 mean proportional, the locus of C is a right 

 line through O, which is of course easily 

 proved." Yours very sincerely, 



A. Cayley. 

 Ca.mbridge, M<ay 5. 



George Bruce Halsted. 

 Austin, Texas, Feb. 15, 1895. 



