[SULLIVAN] A CERTAIN PROJECTIVE CONFIGURATION 

 But by (17) 



163 



dt 



+ ^-^(^°^7t)^^ + ^'^C''7?)' 



hence dv 



Gi{v) = Giiu) = Cl (a constant), 

 The integrals of (21) are readily found to be 



f 



(22) 



^ = Ci 





</> 



= Ci 



dv 

 du 



+ C/i, 



+ Fi, 



where Ui and Vy are such functions of u and v respectively that T 

 and f ' satisfy (17). 



It follows from equations (16) that 



(23) 



Hence 





dît 



du 



ôlog -7=:- 



dv 



(chY'^-hY- 



-^ = r J (^=^du + /^^=^^ ^y I +C2 (a constant), 



where the expression under the integral sign is an exact different ial. 

 Now substitute the values of , — and ^, — found in (22) in this 

 equation; the resulting equation 



(24) 



y 



J 



Cx 



du dv 

 à 



+ 



U\du + 



Vxdv + f2 



is therefore the general integral of the Normal Equacions (6')- 



In order to resolve the general integral (24) into a fundamental 

 set of solutions, it is necessary to express Ui and V\ in terms of the 

 arbitrary functions U and V of equation (12). This can be effected 

 most readily by Wilczynski's procedure,* which we now adopt. If 

 we put 



Uo = 



V. = 



7F ^'' ^' = 



du 

 dv 



sIV 



then, 



*Directrix Curves, Math. Annalen, Dec. 1914, p. 158. 



