3S6 MR. WARREN ON GEOMETRICAL REPRESENTATION 



the second spiral will cut its radii at an angle = k -\- m; 



But«' = zcEy^"^, 



p 



k + m. V- 1 



.-. f = /Ey"(!) ^ 



.•. (by Art. 42 and 43.) f 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, f is a radius vector of the second spiral. 



51. Cor. I.) If /a\'"'^~' = §, and m be a possible quantity; g 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. 



p 



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 g will be a positive 

 quantity. 



53. Let a"* = §, and let a and m be any quantities whatever ; then the 

 values of g are in geometric progression. 



For /aV* represents any one value of g, 



.-. if we substitute (or p successively 0, 1, 2, 3, &c., also — 1, — 2, — 3, &c., 

 we shall obtain all the values of g ; 



Let a! = n, 







then (by Art. 8.) a' = n -{- p c ^ — 1, 



'• ( (o>r \ =nm-\-pcmJ— \, 



