[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 161 



(18-a) U =(-^^ku + h^^h'^' + c^^^k, 



where a^^^ = ^^, b^^^ = -20, ^^^^ -ii^+ .^plogVA%0, , 

 5î« <A \ du / 



ôlog- 





d log V ô log V <j) 



dît dv 



a,(f ) = 0, boS^^ = 0, cP = i ^ _ /aiogVTV . 





and the equations satisfied by f' turn out to be 

 (18-6) r'«» =a ^^'h'u + 6 (f')f'. + c(f') f, 



where 



4> \ du / 



5 log — 7^ , 



ôz> du 



202 _|_ a log V </) _ a log V </) ^ 



ay " ^ 



5 (^'°gy^y + *» 



Therefore the locus P is a plane pencil. 



Four Invariants of the equations (18) are of particular importance 



in the present study; they will be denoted by the symbols H^^\K^^^ 



H^ \K^^ \ and are indeed the Laplace-Darboux Invariants of the 

 equations with mixed derivatives of the second order.* The values 

 of these Invariants are 



*N, pp. 492. 495, 476. 



