AND LINES IN THE INTERSECTIONS Ob’ A SYSTEM OF SURFACES. 
203 
Then, from (97) 
[f] (Sf) + M (3 tj) + [f] (Sf) 
r K f] (Sf) 3 + [v, y] (Sy) 1 + [f, f] (Sf) 2 -I 
5 l + 2 [y, a (Sy) (SO + 2 [o f] (SO (Sf) + 2 [f, ,] (Sf) (S,) J 
-}- terms of the third and higher orders = 0 . .(98), 
and by means of the substitutions in (II), (25), (26), 
/(£ v> £> P) 
+ [f] (Sf) + M (§0 + ra (SO + [a] (Sa) + [0 (S/3) 
[f> f] (Sf) 3 + fo, 0 (S# + ft a (SO 3 
4- 2 [ v , 0 ( 8 ,) (SO 4- 2 [0 £] (SO (SO + 2 [0 o (SO (SO 
+ 2 [0 a] (SO (SO 4 2 [77, a] (S77) (Sa) + 2 [0 a] (SO (8a) 
+ 2 [0 0 (SO (h0) + 2 [ 77 , (3] (SO ( 878 ) + 2 [0 0] (SO (S/3) 
+ [a, a] (Sa) 3 + 2 [a, /3] (8a) (S/3) + [/3, 0] (S/3) 3 
+ terms of the third and higher orders = 0 (99), 
[«] + [a, O (SO 4 [a, O (SO + [a, 0 (SO + [«, *] (Sa) + [a, 0 (S/3) 
4- terms of the second and higher orders = 0 . 
( 100 ), 
C/3] + C/3, 0 (SO 4- C/3, 0 (SO + [A 0 (SO 4- [A «] (SO 4 [A /3] (S/3) 
4- terms of the second and higher orders — 0 . . . . (101). 
Making use of (11), (25), (26), (98), equations (99)—(101) become 
[f, “] (Sf) (Sot) + [y, ot] (Sy) (Soc-) + [f, 01 ] (Sf) (Sot) 
4- [f, A] (Sf) (S 13) + [,, /3] (Sy) (8/3) + [£, /3] (Sf) (S/3) 
+ i [a, a] (Sot) 2 + [ot, /3] (Sot) (S/3) + I [A /3] (S/3) 2 
4- terms of the third and higher orders = 0 (102), 
[«. f] (Sf) + [«, y] (Sy) + [a, f] (Sf) + [ot, «] (Sa) + [a, 0] (S/3) 
4- terms of the second and higher orders — 0 . (103), 
[A f ] (Sf) + [A y] (Sy) + [A f] (S() + [A a] (Sa) + [A /3] (S/3) 
4 terms of the second and higher orders = 0 . (104). 
2 d 2 
