426 Mr DAVIES on the Equations of Loci 



Now, this being symmetrical in respect to 2 a rp and 0, shews that the 

 same angle is formed whether the radius-vector be or 2a 0. That is, 

 at M or M'. Now, referred to the other focus, P'M (= 2 a qp 0) becomes 

 ; or, in other words, P'M = PAT, and . . P'ML = PM'L" = PML. 



Again, put cos 2 a cos 2 e = 2 cos* a 2 cos 2 e, then (3.) becomes 



=5 s T Turn cos2 a cosZ e /-/i 



sin- 1 LMP = -- . .,. __ . . ; -r- ................................. (4. 



n sin (2a_f_0) sm(j> 



which is analogous to a well known property of plane conic sections. 

 Moreover, 



sin PL = sin PM sin PML 

 sin P'L' = sin P'M sin P'ML' 



which, multiplied, give 



sin PL sin FL' = sin PM sin P'M sin" PML 



^ fn ^.\ COS 2 - COS 2 6 



= _i_ sin sm (2 a zr: <p ) . 77= i, . 



' 



= + {cos 2 a cos 2 f } .............................. (5.) 



= db (sin 2 a sin 2 6) .............................. (6.) 



In plane conic sections it is PL . P'L' a 2 (1 e 2 ) = (conj. axis) 2 . 



It would be easy, obviously, to pursue these researches to any extent* ha- 

 ving so many properties of the plane conic sections before us to suggest the 

 correlative spherical ones : this I conceive is unnecessary in the present 

 paper. It will, however, be done in the Mathematical Repository. 



P. 361, 1. 7, dele also 



