128 KEPORT — 1900. 



Also, if any two independent curves of the triplj'-infinite system can be 

 passed through this arbitrary point z and through certain other two x, y, 

 ony=0, then a singly-infinite number passes through these three points, 

 and they form one of the triplets whose number is required. To arrive at 

 two independent curves of the system, take any two fixed points A, B, in 

 the plane, and consider first the curve through sand A, then that through 

 ;:; and B ; each has still one degree of freedom, but loses this and becomes 

 perfectly determinate if passed through a common point y on /=0. Let 

 every curve of the triply-infinite system have M movable points of 

 intersection with /=0 — that is to say, M points whose co-ordinates depend 

 on the variable parameters of the system — then these two independent 

 curves determined by y have each M — 2 points of intersection x with/=0 

 besides z and y ; and in general the M — 2 a;'s belonging to one curve will 

 be all distinct from those belonging to the other ; but if y be properly 

 chosen (or, we may say, for certain positions of y on y= 0) an x of one curve 

 will coincide with an x of the other, x, y. z thus forming one of the required 

 triplets, since two independent curves pass through them. The expression 

 for certain positions ofj oni=0 introduces the idea of the movement of the 

 point y on/, which necessitates a corresponding movement of the two sets of ' 

 M — 2 .r's belonging to the two distinct curves ; we may say with reference 

 to each curve, that to every position of y there correspond M — 2 positions 

 of X ; moreover, since, when confining the attention to the curve through 

 z and A, it is immaterial which of the M — 1 points is called y, we say 

 that to every position of x there correspond M — 2 positions oi y : we 

 have a symmetrical correspondence (M — 2, M. — 2)heliveen the ^yoints x, j 

 established on f=0 by means of the C7trve through z a7id A ; and, similarly, 

 we have another established by means of the curve through z and B. 

 But we have already pointed out that for certain positions of y, i.e. of the 

 one of the M — 1 points which is common to both curves, there will be an 

 X common to them as well, and it is the number of such positions of y (or 

 of this x, since the relation of this particular x and y is reversible) that 

 we wish to find. 



Again, since the original system has three degrees of freedom, the 

 system through A (or through B) has two degrees of freedom ; hence one 

 curve of the system can always be drawn to touch /=0 at any point on it, 

 and no curve can have a double point. In other words, wherever z is 

 taken ony=0, there is always one position of y which coincides with z (on 

 the curve touchingy=0 at z), or 2/=~ satisfies the correspondence equation 

 ^ identically once. (If no curve could have been drawn to touch y=0 at 

 any arbitrary point, y=^z would not satisfy the correspondence equation 

 identically at all, whereas, if a curve with a double point at any arbitrary 

 point on/"=0 could have been drawn, then y=z would have satisfied the 

 correspondence equation identically twice, &c.) The number of times 

 that y'=z satisfies the correspondence equation is called the ' Wertigkeit' 

 of the correspondence, and is denoted by \<f\,j-. In symmetrical cor- 

 respondences, such as the one above, x, y, z are all interchangeable, and 

 therefore [</)]i„= [</>].„= [()!)],-„._ 



The value of the ' Wertigkeit ' is written as a subscript to the bracket 

 (i^^') ; thus, in the language of correspondences, the number of solutions 

 CO our present problem is the number of pairs of points which satisfy two 

 correspondence equations, each given as (M — 2, M — 2)j. But from this 

 number must be subtracted those pairs of points which lie on that one curve 

 of the triply-infinite system which passes through both A and B as well as 

 z, for these do not lie on two distinct curves, and therefore not on a curve 



