62 Professor Hamitton’s Third Supplement 
from which relations, combined with (D*) (#’*), and with the equations (B) (#), of 
the second number, we obtain the following generalisations of the equations (B), 
ov ob or Vv — Vo 
Sz, ~8a,> &y, — §8,> &, ~ &,’ WI 
sv _ ou vr <A ee: ~ 
tae Se eee oe 
and therefore the following aval ea of the fundamental formula (4), 
év bu’ ron Sy? bv og bu > : 
sV= 5 cna pi? = 3a, Fi ox, sf ap, % oy, — a +5 ae aed re 3y/ 8z, ’ (1°) 
which is thus shown to extend to oblique co-ordinates, < not even to require that 
the initial should coincide with the final axes. 
We may adapt nearly all the foregoing reasonings and results, of the present Sup- 
plement, to this more general view. We have, for example, partial differential equa- 
tions of the first order in V, analogous to the equations (C’), and of the form 
OL OV Oa 
0=0 SE AS DSR arc AU. ? 
/ ( Se, Sy een? rentice x) (K*) 
1/80 = SF BA oF 
0=Q (-S, aye Sz’? Se RAED x)» 
which conduct to a partial differential equation of the second order, analogous to 
(D): and if we put the equations (A*) under the form 
0=2, (s,, OP) BY,» > x) 
’ , ‘ , , , ‘ (L') 
0=Q) (c;; COE) 29 Y> Zs x)s 
and suppose them so prepared, by the method indicated in the second number, that 
the function Q +1 shall be homogeneous of the first dimension with respect to o,,7,,v,, 
and that Q,/ +1 shall be homogeneous of the same dimension with respect to o/’, 7/, v,, 
we shall have 
eo (wr) 
with many other relations, analogous to those of the second number. The differen- 
tial equations of a curved ray, ordinary or extraordinary, in the third number, may 
be generalised as follows, 
d §&  & d & =e, d & _w& 
ds Sa, dx,* ds 3B, dy, ° ds dy, 82,’ 
(N*) 
and their integrals may be extended to ee co-ordinates, under the form, 
