. 84 



tions are of such a nature as to render integrable the equation (E), the integral will represent an 

 infinite number of different mirrors, each of which will possess the property of reflecting to the 

 given focus, the rays of the given system, and which for that reason I shall call focal mirrors. 

 [7.] To find under what circumstances the equation (E) is integrable, I observe that the part 



eidx + fi:'dy -If ^ dz 



is always an exact differential ; for if we represent by X', Y, Z' the coordinates of the given 

 focus, and by f' the distance of that focus from the point of incidence, we shall have the 

 equations 



X'- X = «v. Y'— ^ = /3'?', Z' — « = yj', 



and therefore 



»'dx •\- ^'dy + ydz = — d^' 

 because i-^'^i^O 



.-s + ^« + ys = 1, cc'd»' + 0d^' + y'dy) = 0. ''- 



If therefore the equation (E) be integrable, that is, if it can be satisfied by any unknown re- 

 lation between x, y, z, it is necessary that in establishing this unknown relation between those 

 three variables, the part (u.dx + /i.dy + y,dz) should also be an exact differential of a function 

 of the two variables which remain independent ; the condition of this circumstance is here 



<-*-'e-f)+'^+«(|-S)+<»+^(|-t)=°. m " 



and I am going to shew, from the relations which exist between the functions », /3, y, that this 

 condition cannot be satisfied, unless we have separately 



d^ £l_o ^^——0 — _ — =0 (G) 



dz dy ^ dx dz ^ dy dx 



that is, unless the formula {ct,.dx + ^.dy ■\- y.dz) be an exact differential of a function of x, y, z, 

 considered as three independent variables. 



[8]. For this purpose I observe, that since the functions «, /3, y, are the cosines of the an- 

 gles which the incident ray passing through the point (x,y, z) makes with the axes, they will not 

 vary when the coordinates x, y, z, receive any variations ix, 3y, Sz proportional to those cosines 

 «, /3, y ; because then the point x-^-^x, y-\-^y, z + h, will belong to the same incident ray as 

 the point x, y, z. This remark gives us the following equations. 



