E\ m a/ — 1 



OF ALGEBRAIC QUANTITIES. 355 



.•. X b' +a?.a+ V^'s/ ~ ^ =wx /'E\"' = w x cos m + « a:' sin w . ^ — \, 



.*. J? . a + jp c = w X sin »j, 

 but n X sin m is the angle at which § (considered as the radius vector of the 

 spiral) is inclined to unity. 



.-. X . a, + p CIS, the angle at which f, as radius vector of the spiral, is in- 

 clined to unity ; 



but when x =■ \, ^ ■= a and angle x . a, •{• p c becomes a -\- p c, 

 and a is less than c, 



.-. g becomes equal to a in its p + l*"" revolution. 



48. Cor. 3.) If a be positive but not = 1, and p = 0, the spiral becomes a 

 straight line ; if a = 1 , and p be not = 0, the spiral becomes a circle ; and if 

 a = I, and p = 0, the spiral becomes a point. 



49. Cor. 4.) If a be any quantity whatever, and «' = n /EN"* "^ ~^,n being 



positive and m possible ; and if a logarithmic spiral be described having its 

 pole in the origin of a, and cutting the positive direction at a distance = 1 and 

 passing through the extremity of a in its/? +1"' revolution; then the spiral 

 will cut its radii at an angle = m. 



50. Let (a\ yo) = f, where a is any quantity whatever, and m 



any possible quantity, and n any positive quantity ; and let a logarithmic spiral 

 be described having its pole in the origin of a and §, and cutting the positive 

 direction at a distance = 1, and passing through the extremity of a in its 

 p _). 1th revolution; and let a second logarithmic spiral be described having , 

 the same pole with the first spiral, and also cutting the positive direction at a 

 distance = 1, and cutting the first spiral at an angle = m; then § will be a 

 radius vector of the second spiral. 



For let a' = I /E\* '^~\ where / is positive and k possible, 



then (by Art. 49.) the first spiral will cut its radii at an angle = k, 

 and since the second spiral cuts the first at an angle = m, 



2 z2 



