48 Professor Hamitton’s Third Supplement 
referring to the variations of « y z y, and to combine the four thus altered with the 
three following, 
Cagis é ly sQ_ SQ .o SQ .o&v FQ ww 
v 
l 
"®a = So8e” Bat Sede 3B * Sede By? 
v SQ SQ .& SO .&v SQ .& 3 
Jou, stay ee a | Ory $38 + Say ° By’ (1) 
1 bv 82 82 .& FO .&v FQ .& 
2 ee 813 Babs ° ba t Brdz 5G t Bude ° By’ 
in which ¢,, is the same new characteristic, and which are deduced from the equations 
already established for variable media, 
1o 00 
de or? 
and we are conducted to a formula for 8v, which no otherwise differs from (H°) 
than by having 6) instead of © throughout. 
And if, reciprocally, we would express the law of dependance oF the coefficients of 
Q of the second order, on those of v, we may do so by the following general formula, 
v'v'(v8Q +820 +:28,080) = ot v (=, 2 ai (aa) b 
+ oe ( eg 2, &) — » (20— a5) Y 
feck 3) Bee: 3,53) ¢ 
+25 » (a=, 5)—+ (a-2,5 =) ES «( (8- 3) — ee. a5) } 
Pee x= a, =) — » ( do—8, = haa Be 8,) — o ( ar— -8,53)} 
+ oF 4 (8,2) = 6(%=8 “tye (8r— 8, + (8-3, 2) b; (KH 
in which 6, refers still to the variations x, y, z, x, and in which v” has the same mean- 
ing as in the First Supplement, namely 
Sy oy? ve oer 
»_ ov eal So \? sv Sv sv_\? Susy sv \*_ ; 
= 59 3p ~ (Sax) +5pr 3° ~ (Say) top ae — 4) ee) 
this quantity v” is also connected with the w” of (G*) (7°), by the relation 
2 2 2 
vy" On = ott tu . (M’) 
vt 
The formula (A) is equivalent to twenty-eight separate expressions for the partial 
differential coefficients of Q, of the second order, which extend to variable as well as 
