712 
MR. A. McAULAY ON THE MATHEMATICAL 
and, therefore, 
sS'Hx“ lcl), x ,_1 £ = y Vx£x -l4> Y -1 £ + SxYx£x~ 1< hY~ 1 £ 
= 2SS x £3 ) Y“ 1 £ [ibid., eq. (6)] 
and Syw = — SojV. Bp [ibid., eq. (25)]. Hence [ibid., eq. (7)] 
j j f<9S (x/6)ds = - fffsSp'^V-’V^s' 
= — [[|SSp/'NV/c/?' [ibid., eq. (27)] 
= - ffs Bp'&dZ + (([sSp'^V/ds' [eq. (4), § 5, above]. 
Hence, from equations (24) (25), above, 
P = ®'A' , F, = - [*'U„3, + 1 .(34), 
showing [ibid., p. 107] that the presence in x of 4' leads to a stress <t>. 
40. Now, suppose the only variation of eq. (24) is that of temperature. In this 
case 
Y \+ ) = 4 r Y- 8 89 ^ 
= - ' Be - SV B9qVx, 
u 
since [Routh’s ‘ El. Rig. Dyn./ 4th ed., § 410], 0 (x + £)/B9 — 0. Also [§ 5, eq. (4) 
above] 
- ( jjsv B9 e Vx ds= - [ ( Be S d$ & Vx + |[[ 8^SV 0 Va: c ?5 
Hence, the variation of 9 leads to 
j = (x + f ye - sveVx .(35), 
f, = [SU^V®]. +J .(36). 
41. The first of the three statements in § 37 is now obvious, as far as c is con¬ 
cerned. With regard to C it must be remembered that C cannot be made to vary 
without varying H. Now [Rayleigh’s ‘ Sound,’ 1st ed., I., § 81] in order that frictional 
forces may be explained by a dissipation function X, in Lord Rayleigh’s sense, the 
frictional force Q corresponding to an independent coordinate q should be = — B'X./dq. 
For our purposes this is put more conveniently by saying that 2Q Bq = — X B'X, 
