152 THE ROYAL SOCIETY OF CANADA 



In this paper certain new metrical properties of the linear com- 

 plexes involved are also placed in evidence. These properties may be 

 stated as follows: 



If C'(v) and C"(u) be the linear complexes determined by the 

 asymptotic curves (z; = const, and m = const, respectively) of a surface 

 whose Directrix Quadric consists of two coincident planes, the loci 

 of the axes of C'{v) and C"{u) are two cylinders with parallel genera- 

 tors; and the principal parameters of C'{v) and C"{u) have the same 

 constant numerical value. 



Analytical basis of the projective differential geometry of non-developable 



curved surfaces. 



The projective theory of non-developable curved surfaces, as 

 developed by Wilczynski, is based on a discussion of a system of 

 partial differential equations which can be reduced to the form 



P^ + 2a(u, v)^ + 2 b{u, v)^^ + c (u, v) y = 0, 

 au^ au ov 



(1) 



^ + 2a'{u,v)^-h2b'{u,v)^-hc\u,v)y = 0; 

 dy ou ov 



and on the theory of the Invariants and Covariants of this system of 

 equations. To make this clear it is first necessary to outline briefly 

 Wilczynski's procedure. 



If we differentiate equations (1) with respect to ti and v, we find 

 four distinct third order derivatives ; and each of these can be expressed 

 uniquely in the form 



dy dy d'^y 



a ill, v)y + ^ ill, v)^-\- y{u,v) ^+b {u, v) 



du ^ ' ' dv ^ ' ' dudv 



By a further differentiation we find that each of the mixed derivatives 

 of the fourth order can be calculated in two distinct ways. The two 

 expressions thus obtained for each of these mixed derivatives of the 

 fourth order must be identical; hence the coefificients of equations 

 (1) must satisfy the following integrability conditions: 



Of, = 0, b'u = 0, 

 (2) a'uu + c'u — 2a' Ou — 2aa'u — («ro + 2b' a^ — 2ha\ — ia'b^ ) = 0, 

 b,, -{- c, - 2b, b' - 2bb\ - {b'uu + 2ab'u - 2a' bu - ^a'u b) =0, 

 c'uu- 4:ca'u - 2a'cu + 2ac'u - {c,, - ic'b, - 2bc', -f 2b' c, ) = 0, 



where 



da _ da _ «9"o 



au — ^ } a^) — ^ , Ouu — -1.2 ' ^tc, 



and when these conditions are satisfied the system (1) is completely 

 integrable. 



