On Systems of Rays. 23 
os OV i i 
nine of the remaining ten, namely all except 52? We may obtain expressions for 
x 
these by differentiating the three equations (G’), and comparing the differentials with 
(4°) ; for thus we find, 
Sv BW EW eV SW OV, 
Sc ~*~ —Sad2’ Sxdz' Fro’ Sydx"’ 
SY SW eW eV ew SV 
S77 Sy? Sadi Sxdy drdy Syd? 
Sv SW ewer Ew RV 
S22 8e* Bade! Bde BrdeY BySY ? : 
SF BW OW BV MW (L’) 
Sx/dy’— Ox’Sy’ Sad’ Sxdx/ Sex" Syd’ ? 
SV BW Ww BV ew eV 
3/82 —«By/Sz’— Baby’ Sxd82 Sd Bydz’” 
OV ¥ OW SW 82V CW 8V : 
32/82’ —Ss'Sx" Sade’ Broa’ drdz’ Syda’ ” 
and 
SV BW. Bw BY Bw eV 
Sx'dy SO aw'By Sada’ Ardy —Srdx’ Bydy ” 
A lL ol Al z 
By 8x = ByBx  Baky” Rady — BrBy” ByBy” ae) 
yy Pe ke On ae. eo Reel 
d2'dx ~ 82'8x ~ 8082’ oudy ~ 8482! aydx - 
the equations (G’) give also 
SV BW BW BV BW BV 
&x'8y——C'Sy—Sady’ Bxd2’ Bry’: By/Sz’ ; | 
OV a OCW SW 8V OW §V 5 
dy7d2'——«Sy'S2—Sadz’ Bx8y/ Srdz’ Sydy' ” (N°) 
o2V CW SW bW OoW BV 
378n'870x' Sada’ Sxdz" Stn’ yds? 
but these three expressions (V*) agree with the corresponding expressions (L’), 
because, by (J*), 
ew ey SW ey ew ey ew eV 
Badu’ Sudy/ * Sedu Sydy’ — Sady/ Sede! * Srdy” Bye"? 
OW SV OE CRUE ORG IZ OW SV 
Body’ Sede’ * Srdy/ Sydz’ Sade’ Sedy’ | Sede’ dydy’? 
Deige OA Van Ong 10202 OV ORV 987 Wa Oi 
SaBz’ Sede’ | SS? SySe’ ~ Sade’ Sude’ * Srda dy8z'* 
(0*) 
2 
Finally, with respect to the twenty-eighth coefficient , this may be obtained by 
differentiating the third equation (B*), which gives 
