64 Professor Hamitton’s Third Supplement 
7 Ges Ga mans 
these new equations (7'*) (U*) being analogous to (J*) and (P”). It is evident that 
the arbitrary constants introduced by these transformations of co-ordinates must often 
assist to simplify the solution of optical problems. In the comparison, for example, 
of a given polygon ray, ordinary or extraordinary, of any given system, with other 
near rays of the same system, it will often be found convenient to choose the final 
portion of the given polygon ray for the axis of z,, and the initial portion for the 
the axis of z/, a choice which will make a, 8, a/ 3’ and many of the new partial dif- 
ferential coefficients vanish, without producing, by this simplification, any real loss of 
generality. 
We may even carry these transformations farther, and introduce polar co-ordinates, 
or any other marks of initial and final position, and still obtain results having much 
analogy to the foregoing. For if we suppose that the final co-ordinates 2, y, 2 are 
functions of any three quantities o, 0, ¢, and that in like manner the initial co-ordinates 
x’, y’, z' are functions of any other three quantities o’, 0’, ¢’, so that 
87, 
z, 
ar a5 tp +5 +5 80 + 8, dx= = ee +o dé c dd, 
_ oy oy oy 
oy = se 8p +55 00+ Z 36, dy= 3p dp + 350 +3 L dp, 
pum, + 59 i d x = a0 += ad 
7 = 3? "30 mae Bs 25 Oats ap OP “ 
Bd ee Selg ag a 7 alae Borg ela preaclany:+ | oe 
on ~ 80! +o + > ae 0! Pt Ay 39" P » 
ete seb DEV ne penny A 12 By 
oy = 30 do + sq 80! + = dy The dp + aqr 28 + 3g" dp’, 
heise ale) Nabtes slay Vg 
oz Saar + 397 8 + =e dz'= aaa + 30 do! + 9 dp, 
we ee consider V as a function of o 0 : 0 ¢' x, obtained by ee for vy Z 
x y 2 their values; and if we substitute also the values of dx, dy, dz, in the differ- 
ential dV’, or vds, which was before a homogeneous function of the first dimension of 
dx, dy, dz, such that by our fundamental formula 
dV duds_ ov _ 8V 7 
édzx ddx ba 
8dV _bds_ dv _0V \ 
ddy ~ ddy 8 by” Se 
odV _ dvds _ dv _8V 
Sdz ~ 8dz by bz’ 
