350 Mr DA VIES on the Equations of Loci 



in the last case) ; PM = <, RPM = 6, and RMT = L, we shall have 



, cot a sin cos (b cos 6 , cot a sin d> + cos d> cos & T ,_ . 



cot- 1 -*V-a z +COI- 1 z^-fi -*- -=L (8.) 



sin u sin 



or, taking cotangents and reducing, we get 



{cosec 2 a-sin 2 a} S my-l =cotL 

 2 cot a sin sin 



or, resolved for 0, this gives 



cot L cot a sin 6 -+- {cosec 2 a 1 cot* a cot 2 L . sin* 6}* 

 sin(b= ==-* i ^-5-3 *- (10.) 



' rnspp z ft. sin* ti 



cosec 



3tio, Take the same origin of (j>, but refer 6 to a meridian PQ at right 

 angles to RT, then, we have merely to interchange cos d, sin 6, and we get 



cot L cot a cos & {cosec 2 a 1 cot 2 a cot 2 L . cos 2 Q] 2 

 sma> = - 5 - 5-3 - - 



cosec 2 a cos 2 a 



, If, on the contrary, we take rectangular co-ordinates, P being still 

 the origin, we shall have 



SK Q. 



L =r cot" 1 cot a + 6 sin (f) + cot" 1 cot a 6 sin <^> 

 or, taking the co-tangents of both sides, 



sin 2 (f> cot a + B cot a- 6 1 _ 



_ t 



sin <J> (cot a + Q + cot a 0) 



and, by putting for these co-tangents, their values, in terms of the sines, &c. 

 it is 



1 



cotL singarirlcot'L sin* 2 a sin 2 2 



