720 
MR. A. McAULAY ON THE MATHEMATICAL 
- jjjySV SD ch - (jy,S dt SD - jj]SA(SC - VSH/4 tt) ds + ^tt)" 1 ff SA, dt SH. 
where 
[2/Ja = [jZjij [Aja — [Aji, 
and where y, y s are scalars and A, A^, vectors ; SD and SH may then both be regarded 
as arbitrary. The expression to be added to SL may, by equation (4) § 5, be written :— 
- | jjjSA SD S? + j]j{S SD (A + Vy) + S SHVA/4 tt} d? 
- jj{(y + 2/,)SSDSS- SSH(A + k s )dtli7r] . . . (16). 
Equating now to zero, the coefficient of SH in the extended SL, we get 
B = VVA, [V dtA] a+i = 0, 
the A s disappearing on account of the relation [AJ 0 = [AJ 4 . This is the promised 
proof of equations (19) (20) of §26, and, therefore, also of equation (18) of the 
same article. 
49. SL and 2Q Sy are, [(9), (15), (16)], now in such a form that the consequences of 
equation (l), § 13, are seen by inspection. They give (writing D' m for D m /m, so that 
D',„ is the density of matter in the present position) 
D>' = - D'.V'W + (f + 4» , ) 1 V' 1 + F.(17), 
b — — [(<£ + ) Ur ]«+i + F,.( 18 ), 
e = d VZ .. (19), 
E = D VZ — d c Vl/di — A — Vy .(20), 
e. f = 0.(21), 
E, = [yTJvja + b .(22). 
Let, for a finite region 
P — rate of doing work of external forces 
-f- rate of supply of heat from external sources 
(23), 
