128 



and that the sign of that quantity {F") which distinguishes between the two different kinds of 

 aberration from a principal focus, and which, as we have seen, depends on the number of di- 

 rections in which the perpendicular surface is cut by the osculating sphere, depends also on the 

 number of directions in which the mirror is cut by its osculating focal surface. 



To prove these theorems, I observe that if we denote by (p", g") the partial differential coeffi. 

 cients, first order, of the focal surface, that is, of the surface which would reflect accurately the 

 rays of the given incident system to the focus (X, Y, Z), the condition that determines the di- 

 rections, in which this surface cuts the given mirror, with which, (by Section X.) it has com- 

 plete contact of the second order, is 



d^p'\diic + (^fJy = (Ppdx + ^g.dt/, (H") 



and that this same equation, when it is to be satisfied independently of the ratio between dx, dy, 

 resolves itself into four distinct equations, which are the conditions for contact of the third 

 order, between the given mirror and its osculating focal surface. Now, if we represent by 

 «"> /3"> y", the cosines of the angles which the reflected ray would make with the axes, if it 

 came from the focal surface, and not from the given mirror, we shall have (Section II.) 



«" + «' + (V" + y')/' = 0, ^"+^'+(y"+y')/' = 0, 



and therefor^ 



d»"Jrd«i-\-[dy"J^dy). p'+(y"+ vO- df> = 0, 



rf^"+cf/3'+(rfy''+rfy'). ?"+ (y" + -A' dq" = 0, 



d'<^+d'»' + {d^'/ + d^'/}. /+2(iy"+rfy'). df+ (y'+y'). d'p" = 0, 



d^fi"+d''fi' + {dV+d^V)-f + ^id'/ + dY'). dq"-\-('y"+Y')-dY = 0, 



«', ,3', y, being the cosines of the angles which the incident ray makes with the axes ; in the 

 same manner, we have for the given mirror, 



u + »'4-(.y + y')-p=0, /3 + /3' + (y-l-y')-? = 0, 



d» + dcc' + (dy + dv')p+(y+y)dpz=0 



d/3 +d^' + {dy + «?/)? + (y + y')dq = 



d^cc + dW + (c^y + cPyOi) + 2(rfy + dy')dp + (y + y')rf^p =s 



rf^^a +. rf*/3' + {d^y + dS')q + 2(rfy + "'v' )<^? + (v + y^d^q = 0. 



and, because Of the contract of the second order, which exists between the two surfaces, we 



have 



f = p, q" = q, «" = w, /3" = /3, y" = y, 

 dp"=dp, dq"=dq, d»"=d», dfi"zzd^, dy"=dy, 



(y 4- y')[d^p"Jx + d^^'.dx) + 2(<?y + dy')[dp.dx + dq.dy) 

 + <^(«" + »). dx^ d\^"-\-li'). dy + rf2(y" ^ y). c?2 = 0, 



