52 



duct to the same pair of natural coordinate planes as those which we 

 have deduced in the foregoing number, and because they furnish new 

 applications of the characteristic functions of the system. By a virtual 

 focus of a given ray, we shall understand a point in which it is nearest 

 to an infinitely near ray of the system. To explain this more fully, 

 let us observe, that if we establish any arbitrary relation between 

 a, /3, y, distinct from the relation «^ + /3^ + y* = 1, we shall obtain 

 some corresponding relation between 



ST SK SF 



by eliminating a, /3, y, between the equations (C) ; the result of this 

 elimination, which we may represent by 



F denoting an arbitrary function, will be the equation of a pencil, 

 that is of a surface of right lines, composed by rays of the system : 

 and unless this surface be one of the developable pencils deter- 

 mined in the ninth number, the rays of which it is composed will 

 not intersect consecutively, so that there will be only a virtual inter- 

 section, or nearest approach, even between two infinitely near rays. 

 To find the coordinates of this virtual intersection, we are to seek the 

 minimum of dx" + Sy* + h*, or of Jo/* + 3/* + iz'\ corresponding to 

 given values of a, /3, y, 5a, 5/3, 5y. Now if vfe put r = ux + (Sy -{■ yz, 

 we shall have 



« = X, + *r, y = y^ + fir, z = z, + yr, 

 'ix = 'ix, + 3.<»r, 3y = 'iy, + S./3r, 3z =: 3z, + 3.yr, 



and therefore 



} 



(X'") 



^x! = r3« + 3*^ — « («3j;, + j83y, + y3z,), '\ 

 Y = '■S/3 + 3y, — /3 (<.3x, + /33y, -|- 73z,), > 

 Jz' = r3y + 3z — V («3*, + /83y, + y3z,), 3 



(Y'") 



