14 Professor Hamitton’s Third Supplement 
The first equation (U') has for transformation the second equation (U’), when the 
initial medium is variable; and it has for integral, when the initial medium is uni- 
form, the system (E’) (N’) (0'), by which, in that case, WV is deduced from the 
arbitrary function 7’: while, in the same case, of an initial uniform medium, the 
first equation (U’) becomes of the form 
,foW SW sw 
0=2 Gas i? a? XD (A®) 
and is an integral of the following equation of the second order, analogous to ( Y’), 
SHE OU Ua iy Paar gg BN 
e a oy? Bt aa Syse Wea — 
ev SLANG LLL : 
a5 oe +5 by? = al Tay wap (BY) 
When the final medium is variable, the function /V” satisfies the following partial 
differential equation, analogous to the general equation (D), 
Sn SW, Wing, Sel, CU Mie, By PY 
Sade’ brdy' Sudz' © dady’ Broz’ Svdx’ dadz’ drdu’ dudy’ 
WSU Ce CHE a Sl SW SW i Pale ow OW p Cc) 
~ Suz" Srdy/_ dodz2' + 3y 3rdz' Sodz" ' dvdz’ S782" dady’ ” ( 
and when both the extreme media are variable, the function 7’ satisfies the following 
analogous equation, 
OT OT aT OT oT OT eT BT ET 
ba8c' Srér’ Sudu’ Sadr’ Srdu' Budo’ Sadv’ Orda’ Oudr’ 
LET MG BE go SL ST oT, MT ST AT 
~ Suda’ Srér! dadu’ + Sar! ord! Seda" * Suv! oréa’ dadr’ 
(D*) 
General Deductions and Transformations of the Differential and Integral Equa- 
tions of a Curved or Straight Ray, Ordinary or Extraordinary, by the Auxiliary 
Functions W, T. 
6. The auxiliary functions JV, 7, give new equations for the initial and final por- 
tions of a curved or polygon ray. Thus the function V7 gives generally the following 
equations, between the final quantities c, +, v, analogous to the equations (VY), 
ed =const., = const., = const., (E*) 
in which 2’ »/ 2’ are the co-ordinates of some fixed point on the initial portion, and 
the constants are, by (G'), the corresponding values of the initial quantities o', 7’, v’. 
The equations (H*) have for differentials the following, 
