196 MR. ROBERT RAWSON ON SINudLAR 



y= + i> which satisfy (77), but equations (72) and (73) 

 will affirm the nature of these solutions. Put L=i— x 1 ', 

 N=i— y z ; then the conditional equation (74) is satisfied 

 "by */w= s/ 1— x z . 1— y z , and (72), (73) become 



dv __ 



dv _ 

 dy~~ 



From these two equations 

 therefore the complete primitive is 



c + xy± */ 1— x z . 1— y z = o. . . . (78) 



Hence y= + i and o?=+i are singular solutions of the 

 envelope species. Equation (78) admits of the form 



c z + 2cxy + y z + x z —i = o, . . . (79). 

 from which 



y—— cx± \'i — c z ^i— x z . . . . (80) 



It is seen from (80) that neither (c) nor (x) can exceed 

 unity, for if they do (y) becomes impossible. 



Because (c z — 1) is negative the curves (79) are a system 

 of ellipses (see Todhunter's ' Conies/ third ed. p. 239), 

 whose principal axes and centre are the diagonals, and 

 their intersection, of the square y= +1, x = +1. 



The conditional equation (74) is satisfied also by 

 \/w=y V 1— x z and Vw — x Vi— y z } giving the primi- 

 tives 



C + X Vi — y z + y */i — x z = O, "I 



[ . . • (81) 



c + yVi— x z ±xvi — »/* = c.J 



