70 DNIVEKSITX OF VIRGINIA PUBLICATIONS 



and therefore 



5 I = y =__^^58) 



a{ch-\-bg — af) b{ch-\-af — bg) c{ig-\-fa — ch) -~2abc 



21. With reference to an arbitrarily chosen system of cartesian coordi- 

 nates, the first partial derivatives of s (52) with respect to these coordinates 

 equated to zero give necessary conditions 



= al, + fil^ + 7^3 + Sl^ = 0, 



8x 

 ds 



dy 

 ds 



13m, -\- yni^ -\- Sm^ =^ 0, 



an^ -\- ;8»2 + y"3 + 8M4 = 0, 



(59) 



dz 



the I, m, n's being the direction cosines of the radial coordinates of P. 

 These show that the sum of the projections of the segments a, (3, y, 8 (laid 

 ofE along r^, r^, r^, r^ respectively) on an arbitrary line vanishes. Hence 

 these directed segments are in equilibrium at P, in the same way as are 

 the segments xi\, yr^, zr^, wr^, and as sides must construct a closed quadri- 

 lateral in space. If we solve (59) for the ratios a:/3:y:S there results 



sin u)i : sin wj : sin w^ : sin w^ 

 where wj is the solid angle at P formed by r^, r^, r^, etc. The necessary 

 conditions of maximum or minimum s at a point P of ordinary position 

 are represented by the relations 



xr, wr, zi\ wr, a s ,„„^ 



-^^JL^ = ^^ = ~-^^^ = — xyziv. 60) 



a p y 6 S <T 



Consider P to be at a critical point. Construct the four spheres DBCP, 

 AC DP, etc. The equations of these spheres are respectively 



= xp^ ^(T, = 2/^2 — "'> = 2^3 — o'j = wp^ — a; 

 Pi! Pi> Ps> Pi being the powers of A, B, C, D with respect to the correspond- 

 ing sphere. Their radical planes 



xp-, = yVi = nh = ^vPi, (=0-) (61) 



meet in the point P. Dividing each ratio of (60) by (61) 



i\ r„ 1\ 1\ 1 



ap^ ~ Pp„ ~ yPs~ 8p^ ~ s ' 

 Hence 



S- = a'Pi + P"-P2 + y-Ps + ^-Pi, 



= ^^- (63) 



xyzw 



The problem of finding P and s is solved when the p's are known. 



