243 BOOK OF MATHEMATICAL PROBLEMS. 



1 193. The general differential equations of surfaces, generated 

 by a straight line (1) always parallel to the plane Ix + my + xz = ; 

 ('2) always intersecting the straight line 



- =-= - 

 I ~ m ~ n' 



are respectively, 



(1) (m + nq)'r-2(m + nq)(l + np)s + (1 4- np)*t = ; 



(2) (ly-mx)'(q>r-2pqs+p*t) 



+ 2 (ly mx) (nx Iz) (qr ps) 

 + 2(ly- mx) (ny - mz) (qs -pt) 

 + (nx - Izfr + 2 (nx - Iz) (ny - mz} s + (ny - mz)'t = 0. 



VII. Envelopes. 



1194. The envelope of the plane Zx+my + nz = a; I, m, n, 

 being parameters connected by the relations 



I + m + n = 0, I* + m* + n* = 1 

 is the cylinder 



1195. Find the envelope of the planes 



( 1 ) cos (& + <p) 4- cos (0 o) + - sin {(/ 4- o) = sin (0 <p). 

 a b c 



(2) - cos (0 -<#>) + | (cos ^ + cos <) + - (sin^ + sin<) = 1, 

 both when 0, < are parameters, and when 6 only is a parameter. 



1196. The envelope of the plane 



sin 6 cos</> sin sin < cos 6 ~ 

 is the surface 



