( 770 ) 



by the dispersion of the singular tangents rMiew bitangents, of which 

 V are isohited. So for tliis case too the equation (5) hokls good if 

 we but extend 2'i\ to those higlier singularilics of which the point 

 is imaginary but the tangent is real. 



Points of contact and tangents both real. Now j)oint ajid tangent 

 of both imaginary singularities coincide. So tlie two branches touch each 

 other. This may he an ordinary contact or a liigher ojie, as the 

 PuisEUX-developments of both singularities correspond in tlie tirst terms 

 (which are then real) ; the unequal terms are conjugate imaginarv. 



If t and V denote order and class of each of the siiignlarities and 

 c a number which need not be known more closely, the numbers 

 D and 7' respectively of the coinciding points of intersection and 

 of the coinciding connnon tangents of both branches amount to 



^ (6) 



This ensues from a relation which always exists for two singular 

 branches touching each other between the numbers 7) and T, namely 



T-D = (,?,* + 1) {^* -f 1) -- (X,* -4- 1) (X,* + 1). 

 wdiere ,^1* and {i* denote the inflectional indices, v.* and •/.* the 

 cuspidal indices of both singularities'). This relation was first deduced 

 by Stephen Smith (I.e., p. 167). If according to (4)^) we express 

 the indices in order and class of the singularities, we find 



T -D-- ,; i; - t^ t^, 

 or, as in our case t^ =z t^ = t and i\ = r^ = r, 



T — D— c' — t\ 

 from which ensue the two equations (6). 



If therefore we disperse both singularities in such a way that the 

 singular points and tangents begin to differ, this causes \Knnt and 

 tangent to become imaginary, whilst D new double points and T 

 new bitangents appear. If among these are D" isolated double points 

 and T" isolated bitangents, then 



D" — t i- c' , j 



r"^v -\-c'.\ 



which I shall prove more minutely in my dissertation. 



By the dispersion of the singularities the numbers of the isolated 

 double points and bitangents have become d"-|- /-[-'' resp. t"-}-;--|-c'. 

 So the equation (5) gives 



n + ^r + 2 (r" + + c') + i:' c, = k + x r 2 {Ö' i~ t + c) + r ^. 



So before that dispersion 



(7) 



V See note p. /6/ — 7GÖ. 



