= 0, 



1890.] related to Families of Curves. 63 



Similarly we should obtain 



/*. n^ o ( fdu dv) , y ,dr) f du dv) 



Substituting these values in the equation 



9f dr) + dj[dv = 



dx dx dy dy ' 

 we have 



( <,du dv\ ( du dv\ /f.du dv\ ( du dv 

 {^x- V ^)[ V ^- V dx) + ^Fy- V dy)[ 7) dy- V d] / 

 or 



fdu\* /dv\* (fa\* + fiv Yl_9 i- t9 - ?^ — 

 ^W \<W \dyj W/ I 1.3# 3# ty ty 



Now if w = ± v, this equation becomes 



/duV /dv\* / a l t V + ^V+2i-- ?^-l-0 

 V3ay \3ay V3t// \3y/ ~ \dx dx dy dy\ ~ ' 



\du dv) 2 {du dv) 2 n 



i£±5f + V5r* 



du dv 



dx ~ dx 



or s s- = + %. 



du dv 



dy-dy 



This proves our point. This proof is not identical with Rum- 

 mer's, though modelled on it*. The mistake made by Kummer 

 was to assume that the locus of ultimate intersections necessarily 

 constituted a proper envelope, and thus that the points under 

 discussion were necessarily foci. I have however dwelt at sufficient 

 length upon this point in my former paper, and consequently it 

 will need no further discussion here. 



5. It now only remains to point out a few examples of classes 

 of systems that come under the case we have been discussing. 

 Our first example will be drawn from systems of curves which 

 furnish solutions of the equation 



d 2 u d 2 u _ 2 

 dx 1+ dy 2 ~ CU ' 



* If we express the equation' v?=v 2 in terms of % and -q it becomes £ 2 + rf + £17 = 0, 

 and the loci of ultimate intersections would therefore appear to be wholly imaginary 

 (exception being made of the points spoken of above). Kummer's method of 

 investigation therefore gives rise to the question whether, if a family of algebraical 

 curves have an envelope which is not a set of discrete points, their orthogonal 

 trajectories will necessarily consist of a family of transcendental curves. 



