1897.] Differential Equation of the First Order. 287 



The values satisfying these conditions will occur twice in 

 the oo 2n_m determined by the m + 1 equations (1) and (6) in the 

 2/i + 1 quantities x 1} ... x n , z, p x ... p n . 



Thus if (8) is not a new relation this oo 271-7 " must contain a 

 double oo* 1 *, which will satisfy (7) but not be a solution of the 

 differential equations (1) and (6). Any complete primitive of 

 the system will define an <x> n on the oo 271- " 1 and it will not 

 generally happen that a complete primitive will, in geometrical 

 language, meet the double oo n in the same point on both 

 sheets. 



The system of values of x ly ...x n , z, p 1 ....p n will therefore 

 be common to two complete primitives, which thus have contact 

 with each other, so that the complete primitives of the system (1), 

 (6) have a tac-locus. This is one of the cases of failure. 



The second is that in which the determinants (11) all vanish 

 in virtue of a single relation. This relation will clearly be 

 included in (7), at least as an alternative, and the other alter- 

 native, if any, must be taken in order to find the singular solution 

 of the system. In this case the complete primitive oo n has an 

 x n_1 of cusps, that is of points where x u ... x n , z are all the same 

 for two consecutive points. The cusps of the system (1), (6) will 

 form an x 2n_m_1 within which the equations (7) define an oo 71 . 

 This will generally not satisfy the differential equations. 



The occurrence of a cuspidal cc' 1-1 on the complete primitive 

 ought perhaps to be considered as normal, since generally the 

 vanishing of the determinants (11) will be secured by the 

 vanishing of a factor of the left-hand side of (8)f. 



If we consider a singly infinite series of complete primitives 

 of (1), regarding only the co n+1 (z, x x , ... x n ), each primitive has 

 the cuspidal oo 71-1 as part of its intersection with the consecutive. 

 If we consider a doubly infinite series then each primitive has 

 an oo 71-2 of points on the cuspidal oc n_1 as part of its intersection 

 with two consecutives, and so on. Thus, whatever the value of 

 in (< n) in (6) there will be an oo n of cuspidal values satisfying 

 (1), (6) and (7) algebraically but not generally satisfying (1) and 

 (6) as differential equations. 



§ 10. The condition (8) is satisfied also by the vanishing of 

 any factor that occurs to a higher power than the first in any 



* Possibly the double systems may form a manifoldness of more than n 

 dimensions. In such a case the equations (1), (6), (7) will be satisfied by an 

 arbitrary oo " contained therein, but in general this will not be a solution. 



+ For instance, take the equation z=j) v T 1 +p 2 x. 2 +p } p.,. It bas any number of 

 complete primitive forms, representing different doubly infinite series of developable 

 surfaces, and it is only in exceptional simple cases that a developable surface has 

 not a cuspidal edge. 



