220 CELESTIAL MECHANICS. I. V. 21. 



2?7i "^ — '", '" ^"' ~ — c sin 5 sin ^ + c^sin >J/ cos (p — cos fl 



cos 4^ sin (p) -\- c" (cos 4' cos (p + cos d 



sin 4/ sin ^). 



" If we determine -^ and 5 in such a manner, that sin 6 



c" ■"' 



sinrj^zz— ; — ,-T-, — TTir.t and sin cos -^ 



s/ {cc + cc' + c"c"y n/ (c- + c'" + c''2), 



c 

 whence cos 5= ,, ,. , — 73- — ttov ; we shall have 



df d^ 



consequently the values of c' and c'' will vanish when the 

 plane of x^^^ and y,^^ is thus determined. And there is 

 only one plane which possesses this property : for if there 

 were any other, and x and y were the coordinates, and d 

 and (f> the angles belonging to it, we should have 



riQ (Jj2J , j2 d^ 



^m-iii — '" '" — i^zzc sin 5 cos ^, and 



l.m '^"^^"^^ "^'^-'" =— c sin sin ^ ; 



but since d and c"ziO, by the supposition, for the supposed 

 plane; and since these quantities have been shown to be 

 =0, for the planes of x^^^, z^^^, and y^,^, z^^^, we have sin 5=zO, 

 and the two planes must coincide. The value of 2m 



^"^^"\^"' -" being equal to ^(c^ + c'^ + O whatever 



be the plane of x and y from which it is derived, it follows 

 that this quantity may be deduced equally well from any 

 other system of coordinates, and that the plane of x^^^ and 

 y^j^i determined by it, will always be that which makes this 

 elementary area a maximum ; and since the angle ^ re- 



