MR. WARREN ON GEOMETRICAL REPRESENTATION 
the second spiral will cut its radii at an angle = k + m ; 
p 
p 
(< 
k + m • V — 1 
.-. (by Art. 42 and 43.) § is a radius vector of a logarithmic spiral which cuts 
its radii at an angle = k + m, and cuts the positive direction at a distance = 1, 
that is, is a radius vector of the second spiral. 
5 1 . Cor. 1 .) If m ^ ~ 1 = o, and m be a possible quantity ; will be a 
radius vector in a spiral (described as in the preceding article) which cuts the 
spiral, in which a is, at a right angle. 
52. Cor. 2.) Hence if a be a positive quantity, and p = 0, the spiral, in 
which a is, will become a straight line, and the spiral, in which § is, will be 
perpendicular to it, that is, will be a circle ; but if a = 1, and p be not = 0, 
the spiral, in which a is, will become a circle, and the spiral, in which f is, will 
be perpendicular to it, that is, will be a straight line, and £ will be a positive 
quantity. 
53. Let a 1 = §, and let a and m be any quantities whatever; then the 
values of § are in geometric progression. 
For represents any one value of p, 
.-. if we substitute for p successively 0, 1, 2, 3, &c., also — 1, — 2, — 3, See., 
we shall obtain all the values of £>; 
Let a' = n, 
p 
O 
then (by Art. 8.) a' = n + p c — i, 
p 
m \ l 
er = (?) 
TO / p \ 71 TO + ]) C TO tj — 1 
