z=- 



270 Dr, Hii'st on Equally Attracting Bodies. 



boloiilj whose director through the vertex is the 2:-axis. It will 

 also be noticed that the mutual position of the two hyperboloids 

 is completely defined by saying that ike asymptotic plane, through 

 the common director, to each surface must touch the other in the 

 attracted point. In fact it is well known that the pencil of tan- 

 gent planes, whose axis is the common director, is homographic 

 at once mth the system of their points of contact with the sur- 

 face H, and that formed by their points of contact with the 

 surface H,. Hence, whatever the position of the two surfaces, 

 the points of contact of these planes constitute two homographic 

 systems of points in the common director. But when their posi- 

 tion is such as above defined, that is to say, when the points in 

 each system which correspond, respectively, to the infinitely di- 

 stant points in the other system coincide, it is well known that 

 the two homographic systems of points form a system in involu- 

 tion whose centre is at those coincident points. 



30. The theorem of art. 28, as above modified, leads easily to 

 the solution of the following problem : — Given any two equi- 

 lateral hyperbolic paraboloids (H) and (H,), and a point in 

 the director which passes through the vertex of the first (H), it 

 is required to place the second surface (Hj) so that correspond- 

 ing portions of both surfaces may attract O equally. 



The problem is solved by performing the following three ope- 

 rations on (Hj) : — First. Make one of the directors through the 

 vertex of (Hj) coincide with the director of (H) in which the 

 point is situated. Secondly. Make (H,) tui'n around the 

 common director until its asymptotic plane through that director 

 touches (H) in the point O. Thirdly. Make (Hj) slide along 

 the common director without rotating, until the point of the 

 same in which (H,) is touched by the asymptotic plane of (H) 

 coincides with the point O. 



The magnitude of the rotation \ in the second, and of the 

 displacement 7 in the third operation, may be thus found, Let 

 the equation of (H,) at the end of the first operation be 

 ax + fiy 



replacing x,y,z by a- cos \ + ?/ sin X, y cos X — ,*• sin ^, ^""7* 

 respectively, this equation becomes 



[(« + «i7) cosX — (/3 + ,^17) sinX].r + [(« + « i y)sinX + ()g + /3i7)cosX]y 

 (ajCOsX— /3, sinX)a:+(ai sinX + /3j cosX)r/ ' 



and is that of (H,) after effecting the second and third ope- 

 rations. But in order that (Hj) may, in this new position, 

 attract in the same manner as (H), the last equation must have 

 the form of (Hj) in art. 28 ; whence, to determine X and 7, we 



