TORSE CIRCUMSCRIBED ABOUT TWO QUADRIC SURFACES. 
540] 
521 
the lines P'G, P'T and P'L, P'M are harmonic lines through the point P'. It thus 
appears that in the particular case where the points L, M are the foci of the conic 
U', the line P'G is the normal at the point P'; and we may say in general that 
P'G is the quasi-normal at the point P' of the conic IT. 
A 
Consider now the torse circumscribed about the conics U, U'; the plane PTP' will 
represent any plane, and the line PP' any line of this torse : projecting on the plane 
of U' with the point A as centre of projection, the projection of PP' is the line 
P'G; which, as just seen, is the quasi-normal of the conic U' at the point P'. 
The projection of the cuspidal curve is the envelope of line P'G, which is the 
projection of the generating line PP' of the torse—viz. this envelope is the quasi- 
evolute of the conic U'; which is the theorem in question. 
C. VIII.