On Systems of Rays. 31 
pil ae Kl) 17 8. \ ie RE SV \? &yv &v eV \? 
ig ~i& oy (ay) Tay a (Se) ee ae ae =, . Ke 
4 sty 30) ” 
- vw \ oo or ou 
» having the same meaning as before: @ also referring, as before, to the variations of 
ay zy alone, and V'" having the same meaning as in the First Supplement. In effect- 
ing this elimination, we have attended to the forms of the functions JV, Q, which give 
Ww Ww eae 
o (dat Von) +r (d+ Vde)ty(I+V2 e)=-97; 
oo oa 
we have also employed the Pace (4°), which give, by Gi aH 
SV eV Le oe a OVE BV EV Ps , OO oO 
ne a aC) = bate 30) 3 Srdy Sze x2 Gydz Vat? gpg 8 
ay &V Sa i aT ee ee a thea g OND: ; 
a =e wo 3) Wiis ia ee | OD 
Breer iyPr Oa, Sy ey oF RF _ 8080 
3x2 Gap as 0G =A Szdx Sydz 822 Sxdy ie aeree 
Having thus obtained expressions (# *) for the three variations 
537 er 
80 ” é&r ” bu” 
it only remains to substitute these expressions in the four last equations ( Y*), and so 
to deduce, without any new elimination, the four other variations 
SL LL 
oe?) Oye Oz dx ” 
after which, we shall have immediately the twenty-eight coefficients of JV, of the 
r) f) 3 
second order. ‘The six coefficients, for example, of this order, which are formed by 
differentiating /V” with respect to o, r, v, are expressed by the six following equations, 
deduced from (E*) ; 
SV &Q 1 ,oV Ov Ss | } 
Sao Vien tap (“sm t+ yy ce 5a) ? 
eV = ey le a 2y 
See li Vs + mp” "Se +o Srey Que ae 
Sew goV OV 
a ar Ce = Bor a) : 
PW BOL) oul ae Sp OV. eh ae oe 
Sadr ae SANT” bcbg? Bebe Syde Be? 
a ee hg Pe a EV ey: 
Scie 1 Mie eee © bytest ety} ee ee? 
oo sa algal cua clay 
—_ = —— ee = — : 
dude dude * wey” ( ss bzdx cis by6z ap bxdy a ey dy? 
