406 



Mr DA VIES on the Equations of Loci 



4. To find the equation of a line which passes through a given point 

 r, Q, and makes a given angle a. with a given straight line r = a$ec 6 /3. 



Let EC be the given line, F' the given point, and E'F'C" the line 

 sought. Put its equation in the form, 



But it passes through r, 6, , and hence 



, 7, or cssr l cos6 l 7 



Also, 7 = E'OC' '= /3 + f , (where E = E'F G), and hence 



r=:r,cos(d, /3+ e)sec(0 



which is the required equation. 



Let e = 0, or the lines be parallel : then 



r = r y cos ((9, /?) sec (6 ,8) 



(14.) 

 (15.) 

 (16.) 



(17.) 



7T 



Let e = -j- , or the lines be perpendicular : then 



r = r, sin 6, ft cosec 6 /3 (18.) 



It will be altogether unnecessary to multiply examples of this analogy 

 between plane and spherical equations, as the method of forming all such 



