124 Professor Hamitton’s Third Supplement 
the auxiliary function 77, when made homogeneous of the first dimension with 
respect to o, 7, v, by the equations 
ow 2 ow ea 
gor tee On 55, t Mig =: 
ow SO ow oO * 
or? iin ore S ordu e sy x3 (x ) 
ew P@) ew Bre) 
du? 4 ou 0; dude i Suda e 
of which only three are distinct, and in which V corresponds to the focus: or from 
the function 7’, when expressed as depending on o, 7, ¢, 7’, x, by the following, 
ia ce A le gts 88 Ss | 3S (is yt 
80°? oo* \érd0" dar dada" Srd0 Sr? \Oada" 
i eee 
oo dr" Oa° Orda Srér’ Sadr \ Sade" Srdzr | Badr’ Srdo" or? dade’ Sadr’ 
_ (8S \-1 68S (BS \? SS SS Ss os / 8s \? 
# hia ) aes (sy) a 2 aor Sadr’ S367’ iv 67” ( Sab’ ) } 
m025) oes (ce 2 
50 ae — (a8) » 
in which as before, S= J’—zv+2'v'. 
When the final medium is variable, we may employ the following equations, 
ow ow ow ow ow ow 
do? or ov? Sadr Oréu dvd 
(a) "(ay)" (apa ae 
oa or ou do dr or vu ov So 
= (22 dQ 62 a) ae 
dc 8c or Cy) ku Oz i 
L SW: Sica tel SO. if OM dy Sik We, 
or, — =— 
a? Sc  B? or y® Sv’ af Sadr _ By Srov _ ya Suda | 
CD 
(Z") 
Sy ov bv ov\ —1 
= (a at Bs + 1x) > 
of which only three are distinct, but which are sufficient to determine the principal 
foci and principal rays of a curved system, ordinary or extraordinary, by the aux- 
iliary function /V’, considered as depending on o, 7, v, 2’, y’, 2’, x, in conformity to 
the new view of that function, proposed in the present Supplement. ‘The new func- 
tion Z’ might also be employed for the same purpose, but with somewhat less facility. 
It was remarked, in a former number, that at a point of vergency the general 
linear expressions for the relations of near rays fail; but the more complex expres- 
