374 Prof. Cayley on Differential Equations and Umbilici. 



and if for greater simplicity we write A=l, then the derived 

 equation is 



O = C/ 2 -4BCB' + 4CB' 2 = 0, 



corresponding to the integral equation 



* 2 + 2B* + C = 0. 



Writing the integral equation under the form 



(* + X)(* + Y)=0, 



we have 

 whence also 



2B = X + Y, C = XY, 



2B' = X' + Y', C = XY' + X' Y, 

 and the derived equation becomes 



0=--(X-Y) 2 X'Y'. 

 And if we represent the roots X, Y in the form P + Q^/n, 

 so that P= — B, Q v /Q=^ / B 2 — AC, Q 2 being the greatest 

 square factor of B 2 — AC, then 



(X-Y) 2 =4Cm r,Y'=F±(QVD+^) 



X'Y'=P' 2 - ~ (2Q'D + Q0) 2 ; 



and the derived equation is 



a=-Q 2 {4DP' 2 -(2Q'DH-Qn') 2 }=0. 



If B, C, &c. are functions of the coordinates {x, y), the equa- 

 tion s 2 + 2B# + C = (z an arbitrary constant) represents a series 

 of curves in the plane of xy ; but if we consider z as a coordi- 

 nate, then the equation represents a surface, and the curves 

 in question are the orthogonal projections on the plane of xy of 

 the sections of the surface by the planes parallel to the plane of 

 xy. To fix the ideas, the plane of xy may be taken to be hori- 

 zontal, and the ordinates z vertical. 



Writing the equation in the form 



(* + B) 2 -(B 2 -C)=0, 

 we see that the surface contains upon it the curve £ + B = 0, 

 B 2 ~ C = 0, which is the line of contact with the circumscribed 

 vertical cylinder : such curve may be termed the envelope, or, 

 when this is necessary, the complete envelope. The equation of 

 the surface has however been taken to be (z — P) 2 — Q 2 D =0 

 (viz. it has been assumed that B= — P, B 2 — C = Q 2 □ ) ; the 

 envelope thus breaks up into the curve (s — P = 0, Q=0) taken 

 twice, and the curve ' z — P = 0, 0=0; the former of these is in 



