104 



pression of this integral (0), instead of a?,, y,, «,, we may do so by 

 the known methods, by putting 



X, = (x — x„) COS. XX, + {y — y,i) COS. yx, + (z — 2„) cos. zx, \ 



y,= {x — x,i) COS. xy, + (y — y„) cos. yy, + (z — 3„) cos. zy, V (P) 



2, = (x — a-„) COS. xz, 4- {y - Vii) COS. yz, + (z — z„) cos. zz, ^ 



*'// ^/z ^// being the values oi x y z that belong to the point upon the 

 ray which had been taken for origin. 



Verifications of the foregoing Developments. 



4i. We may verify the form (0) which we have thus found for 

 the integral of (A), by the following condition, resulting from (M), 



d V », d V fi, d V ,._.,. /Q) 



,,,^._ dz/ IX, z, + A d», ij, z, + B d/i, ij. - y '^ * '^ ' 



of which each member is an expression for the cosine y ot the 

 small angle which a near ray makes with the ray that we have 

 taken for the axis of z^ . The condition (Q) may be put under 

 the form 



^•-_(««, + ^/3,) = Vl-*' -^'^ (^) 



in which, by (O), 



d F _i , - ^Z -f /3/' , d(p, 

 rfz, • ft ~ "*" 2 '^ dz, 



+ 2, 



(») 



C '^" (fd<l>,Y+^ d'^, \ d" //d(p,Y+i d'tp, \ ^ 



o] rf»,''lw«J du,dz,) d/3,''\{d,S,J d^,dz,J > 



