70 Transactions of the Royal Canadian Institute 



= 0. 



dd f 



The Laplace transformation defined above reduces to 6' = -rr ^o"* 



dp 



equation (A). Hence, if to (A) we apply this transformation, there 

 results an equation (A') having three solutions di, d'/, 6z connected by 

 the relation 



a' (g) 6,' + b' (g) e,' + C (q) d,' = 0. 



Then by the theorem outlined above, two applications of a Laplace 

 transformation leads to an equation (A") having a vanishing invariant 



L dp dq \ dp dq/J. 



Thus, if either family of the asymptotic curves of (S) be complex 

 curves, it is necessary that the Moutard equation (A) be of range three 

 at most. This condition is also sufficient. Because, if (A) be of range 

 three, we can select three linearly independent solutions di, 62, dz 

 connected by the relation (18). Hence we have the two relations 



(19) aei-^be2 + cds=l, 



dp dp dp 



ddi ,jdd2 . ^ dds ^ 

 a + b + c = 



From these we deduce 

 ddi 



(20) 



\ dp dp y \ dp ^ dp y 



= c\ Bz — Q2 /+ « I ^1 — ^2 ) , 



\ dp dp y \ dp dp / 



{ ^ dQz . ^^A, a/z, ^^3 . dQ2\ 

 \ di> dp / \ dp dp / 



dp \ dp dp / \ dp dp 



If cognizance be taken of equations (1), these become 



(21) ^ = cy,-bzi, 



dp 



ddi 



= azi — cxi, 



dp 



