104 The Rev. J. Challis on the Analytical 



The limiting value of the right-hand side of this equation is 

 now to be found. 



Let the equations of the three lines anp, Pp,pg, referred 

 to the axes A #,, A y A x p be respectively 



Then by known formulas, 



cos <P Ps =Vl + a*+b» COS<APQ= V \+a!*^lJ * 



1 + ma' -\-pd 



and cos <apq = ,— - — , 2 —- W2 — / ., , 2 r =? 

 c a v 1 + a' 2 + b u . v 1 + m l + p l 



The values of m and jo may be found as follows : — Let 

 A « = h, and N« = i. Then because the line a np passes 

 through the points a and w, whose coordinates are 0, 0, h, and 

 l 3 k f 0, respectively, the equations (1.) become 



I 



(4.) 



And because the line P p passes through the point P, whose 

 coordinates are r + 1, 0, 0, the equations (2.) become 



x,= az t +'r + l \ 



*,«**, s (5,) 



Now the coordinate A 5 of the point p is I + r + r r Hence 

 it follows from equations (5.) that the other coordinates of p 



are i/, — — ', and z t = — . These values must satisfy equa- 

 tions (4.), because the line a tip passes through the pointy. 

 By substituting them in those equations, it will be found that 



— — = a ( h 1 J, which is the required value of m; 



and = ~ L , which is the value of p. 



p I + r + rj r 



By substituting these values of m and p in the foregoing 

 expression for cos <ap g, expanding and neglecting powers 

 of r, above the first, and bearing in mind that 1 + a a' + bb' 

 = 0, it will appear that 



a 1 f, (aa'r — firn 



