210 Mr. LAND^.^''s In'vcjl'igailon.of 



ucnts, we have s — cv y, — -— z — , and t ^^a" K -=-^-; 



^g ^g 



m and « denoting the values of s and /, when g ls = jand 

 y — O', and B being put to denote the difference D - C = ^^ - c". 

 If now /S and (5* be put to denote the cofmes of EP and DP 

 to the radius ^, we fhall, fronn what is done above, have 



, d_ — =: —J — ; 



P =r _ = ^'' ~oy y--t_. 



^' + y- + r = a% (3f^+yy + ^l=.Q; 

 ^'/3' + cy -l-^Trr^V-f ^W, and 3^/3/3 + c>^ + J^^irro. 



Drawing AR fo that D^ x fine of BR (hall be = C^^, it is 

 "jery remarkable, that the momentary pole (P) will run round 

 about the point B, or about the point D, in the fpherical furface, 

 according as the initial pole fliall be in the part BCR or DCR. 

 of the faid furface ; that is, according as D/«^ Is jefs or greater 

 than Qa^ : and that, if the. initial pole {^P) be any where in 

 the great circle CR, the momentary pole, keeping in the arc 

 of that circle, will continually approach nearer and nearer to 

 the point C in the furface of the fphere ; but, by what fol- 

 lows, we fhall find that it never can arrive at that point in any 

 finite time ! 



The equation of the track of the pole In the projedion to 

 which we have hitherto referred will, it is nov/ obvious, be 



B , . C^r-D 



z 



y"^ ~— X x~ ■] ~-^ ; X, meafured from the center A upoii 



AD, bein.g=^; andjy, at right angles thereto, =z/G. 



If C be=:o (that is, if c ht — b'), x will be equal to the In* 

 variable quantity m ; the projecled track, a right Vme parallel to 

 AB ; and the track upon the furface of the fpherej a lejer 

 ehcie in a plane parallel to the plane of the great circle BC, 



5 If 



