1915-16.] The “ Geometria Organica” of Colin Maclaurin. 131 
Bern. 
Let Q lie on a line l, and R on C», then, by Cor. 5 of Prop. IV, P 
generates a curve C 3 „ which cuts C m in 3 mn points P 4 , P 2 , . . . P 3mn , to 
which correspond Q x , Q 2 , . . . Q 3r J on l. 
The new curve is cut by l in 3 mn points. .’. etc. 
Cor. 2. 0 2 is a multiple point on the locus of order 2 mn. 
For on 0 1 0 2 describe a segment of a circle containing an angle equal 
to 0-,PQ. It cuts C m in 2 n points A 4 , A 2 , . . . A 2m . The lines 0 4 A cut C n 
in 2 mn points, to each of which corresponds the point 0 2 . .*. etc. 
Cor. 3. If of the vertices P, Q, R one lies on a curve C m , and a second 
on a curve C n , the third generates a curve C 3mn . 
§ 44. Prop. VI. 
Generalisation of Prop. XIV in Part I. 
R 
In the figure OfiPT and 0 2 QT are constant angles. If P lies on G my 
Q on C n , T on C r , then R generates a curve C Amnr - 
Bern. 
Let P and Q lie on straight lines, and let R lie on a line l: then T 
would generate a quartic C 4 cutting C r in 4 r points, to which would corre- 
spond 4 r points R on l ; i.e. R would generate a curve C 4r . 
Next, let P lie on a straight line, Q on C n , and T on C r ; then R lies on a 
curve C 4nr . For let R lie on a line V , P on l , and T o'n C r : then Q would 
generate a curve cutting C rL in 4 nr points, to which correspond 4 nr 
points on V. 
Hence the locus of R would be a C/ nr . 
Finally, let P lie on C m . Let R lie on a line l", Q on C„, and T on C r ; 
when P would generate a curve C 4mr cutting C m in 4 mnr points : and to 
these correspond 4 mnr points on 1". .*. etc. 
Cor. I. Each of the points 03 L, 0 2 is multiple of order 2 mnr. 
§ 45. Prop. VII. 
Generalisation of Prop. XXI of Part I. 
Let 0 1 P 1 P 2 . . . P n _ x Q be a serrate angle, Q0 2 R a constant angle 
