34 Professor Hamitton’s Third Supplement 
ow ow ow 
ed (Q') 
and denote by 8/V,, 8°77, the expressions already found on this particular supposi- 
tion, for the variations of JV”, of the two first orders, so that, for the first order, by 
(G) CP) (WPS 
V 
8, =1286 + yor +Zou+o00u + Toy, +u0z 5 ox —F 8Q, (R‘) 
and, for the second order, 8°//7,= the value of 8°/V assigned by the formula (L*‘) ; 
we may generalise these particular values 6/7, °/V,, by the following relations, 
SV, =8 WV —w ed, 
PIP, =P WV —w 8 Q — ew SQ 
ow wy Owe 5 
+(o3 Te T ee a Wise ) 80%, 
(8) 
in which 8/7, &W, are general expressions, independent of the condition of homo- 
geneity w,=0, and of every other particular supposition respecting the form of W. 
It is, however, here understood that the final medium is uniform, and that in forming 
the variations of the function W, the quantities o, 7, v, x, 2’, y, 2, on which it 
depends, are treated as if they were seven independent variables. 
And if we would deduce expressions, 8W,,, 8° W,,, for the variations of W, of the 
two first orders, on the supposition that W is made, before differentiation, homogene- 
ous of any dimension 7, with respect to o, 7, v, we may put 
aw, 3W Ww 
Batt Higgs hg —nWe=w,, (T’) 
o 
and we shall have the following relations 
SW, =sW—-w, 8, 
2 W, =" W —w, 82 — 28w, 8Q (Uy 
wy Own ow, 
=— +, — NW, )8Q’, 
5 Mighs TP +, — NW, )bQ 
+6 
which include the relations (S*). The general analysis of these homogeneous trans- 
formations is interesting, but we cannot dwell upon it here. 
Deductions of the Coefficients of T from those of W, and reciprocally. 
9. The general principles of investigation, respecting the connexions between the 
partial differential coefficients of the second order, of the characteristic and auxiliary 
