OF ALGEBRAIC QUANTITIES. 
355 
= M ( 
x b' + x .a + pc .,J — 1 = nx ^E^ w ^ ~ 1 = n x cos m + n x sin m . *J — i , 
/. x.a4-j»c = w^ sin w, 
but w ,z sin m is the angle at which g (considered as the radius vector of the 
spiral) is inclined to unity. 
.*. x . a + p c is the angle at which g, as radius vector of the spiral, is in- 
clined to unity ; 
but when x — 1, g = a and angle x . a -f p c becomes a + j o c, 
and a is less than c, 
g becomes equal to a in its jt? -f- 1 th revolution. 
48. Cor. 3.) If a be positive but not = 1, andj9 = 0, the spiral becomes a 
straight line ; if a — l, and p be not = 0, the spiral becomes a circle ; and if 
a = 1, and p — 0, the spiral becomes a point. 
49. Cor. 4.) If a be any quantity whatever, and d = n ^ ~ l , 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 itsp + 1 th revolution; then the spiral 
will cut its radii at an angle = m. 
50. Let^”(o) mV_ 1 = 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 g, and cutting the positive 
direction at a distance = 1, and passing through the extremity of a in its 
p + 1 th 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 g will be a 
radius vector of the second spiral. 
For let d — l ^ ~ 1 , where l 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 z 2 
