TRANSACTIONS OF THE SECTIONS. 19 



An examination of the surfaces described by the motion of a plane parabola of 

 any order with its diameters parallel to a fixed right line showed that the con- 

 dition of a pair of equal roots in the equation of the third order, 



considered as an equation inX, was satisfied by all surfaces traced out by a plane 

 parabola moving parallel to a fixed line and enveloping any ciu-ve in space what- 

 ever. As singular cases, he noticed the spindle made by causing a parabola (whether 

 fixed or of variable size) to rotate roimd any diameter, the ruled surface with a 

 director plane, and developable surfaces. 



He also showed that when three of the roots were equal, the surface necessarily 

 reduced to a plane or a cylinder. 



These results are, however, restricted by the method of generating the surface. 

 In fact, for the case of three equal roots, when the partial difl'erentials of the third 

 order are in continued proportion, Mr. Cayley has shown that the resulting equations 

 can be integrated and that the integration gives a more general result. 



Note by Mr. Cayley, 

 The general integral of the equations 



«_ j3 _y 

 /3 y 8 



can be found, viz. 'L=^ gives r= funct. s, and - = ^ gives s= funct. t. But 



fi y yd 



r= funct. s is integrated as the equation of a developable surface {p instead ot z), 

 A-iz. we have, say, 



p = ax+hy+g {a and ff fimctions of A) and 



Similarly s= fimct. t gives 



q = hx-{-by+f, 



Observe that the constants have been taken so that ^=h, ^ =h; but in order 



dy ilx 

 that h may in the two pairs of equations mean the same function of x, y, we must 



have a' = — = ^, that is 



or writing a = ^h, g=x^) "^^ i^n-y^ 



Cdh . rg'dh 



p=x(\>h^yhJrX^i, q=hx+y \ ^ + \^' 



where 



x(^'h+y-\.xh=0. 



The last equation gives A as a function oi x, y ; and the values of p, q are then 

 such that dz^=-pdx-\-qdy is a complete difierential, so that we obtain z by the 

 integration of this equation. A simple example is 



p=lh'^x—hy, qz= — hx-\-y log h, hx—y=0; 

 that is 



whence 



■i X X 



z-=. - v^ log - — ^ V^. 

 2^ ^x 4^ 



2* 



