1890.] related to Families of Curves. 61 



3. We now pass on to discuss the case of imaginary loci of 

 ultimate intersections. These are given by the condition 



It will only be necessary to consider one of these two cases, 

 as the discussion of the two will be exactly similar, and we will 

 choose the one given by the upper sign. The relation 



dx . dy _ - 

 requires also the existence of the relation 



drj dri 



dx . dy n 

 i.e. — + t ^ = 0, 



07] 07] 



unless the first of these relations makes <f) (x, y) zero or infinite. 

 Thus, with this reservation, we have at all points of an imaginary 

 locus of ultimate intersections 



dx + idy = J^d% + ^d7] + i(^d!; + ^d7])=Q, 



i.e. x + iy = const. 



If we had taken the lower sign we should have had 



x — iy = const. 



Thus we see that if there be any imaginary locus such that 

 the coordinates of all points on it make (j> (x, y) zero or infinite, 

 then it is conceivable that this may be part of the locus of 

 ultimate intersections of one of the families of the system. If 

 not, then the imaginary loci of ultimate intersections, whensoever 

 they exist, are collections of straight lines passing through the 

 circular points at infinity. Each of these imaginary straight lines 

 will pass through one real point, and these points will belong to 

 the set of points characterized above. Further, as we are only 

 considering algebraical curves, for any imaginary point of inter- 

 section {a + ib, c + id) there will be a conjugate point of inter- 

 section (a - ib, c — id). Thus it is evident from the reasoning 

 contained in my former paper that the imaginary straight lines 

 occur in pairs, the two members of each pair being conjugate and 

 intersecting in a real point, which is one of the points in question. 



The reasoning of this article proves only that if imaginary 

 loci of ultimate intersection exist, then with the restriction speci- 



