On Systems of Rays. 19 
6a SW _8T 
&— — (t+ yt+uz)= => 
da é 
SQ'F 5. BO VAR 
es a, (ox +TYy +uz)= Se 
or or’ 
8 a ayes 
z- > (o@ + ry +z) = <§ a2 i 
fee aa): oe oily A aa cy 
Bie (ow +TY +vuZ = 3y"? 
a SOE Mp ii pee te eee 
y— <7 (6@ try tuzZ) = =37, 
/ a 4 Se. if of. 6T 
Ai (ox trefiud) Sis: 
ou v 
The case of prismatic combinations may be treated as in the fourth number. 
General Remarks on the Connexions between the Partial Differential Coefficients of 
the Second Order of the Functions V, IV, T. General Method of investigating 
those Connexions. Deductions of the Coefficients of V’ from those of IV, 
when the Final Medium is uniform. 
7. It is easy to see, from the manner in which the equations of a ray involve the 
partial differential coefficients of the first order, of the functions V7, /V, 7’, that the 
partial differential coefficients of the second order, of the same three functions, must 
present themselves in investigations respecting the geometrical relations between infi- 
nitely near rays of a system; and that therefore it must be useful to know the gene- 
ral connexions between these coefficients of the second order. Connexions of this 
kind, between the coefficients of the second order of the characteristic function /’, 
taken with respect to the final co-ordinates, and those of the auxiliary function JV, 
considered as belonging to a final system of straight rays of a given colour, which 
issued originally from a given luminous point, were investigated in the First Supple- 
ment; but these connexions will now be considered in a more general manner, and 
will be extended to the new auxiliary function 7, which was not introduced before : 
the new investigations will differ also from the former, by making JV” depend on the 
quantities o, 7, v, rather than on a, p, y. 
The general problem of investigating these connexions may be decomposed into 
many particular problems, according to the way in which we pair the functions, and 
according as we suppose the extreme media to be uniform or variable ; but all these 
particular problems may be resolved by attending to the following general principle, 
that the connexions between the partial differential coefficients of the three functions, 
whether of the second or of higher orders, are to be obtained by differentiating and 
