TRANSACTIONS OF SECTION A. 911 
bi t+e,vt+d’,2%, a’, + 0,0 + e/,2? + d',23, a’, +b’ x + c/42? + d’,28, 
’ 1 So 7 asd , fom ae) Li oc / / fee 4 
By +e etd 2? a’+b tev +d'.a5, a’ +02 + 6.07 + d’,r, 
+ pL 
B+ cv +d',2", a’, + bor + c'g2? + d’.05, a4 + b’ yu + c/,27 +d’ yx, 
Where p does not involve (x): by repeating the process we obtain two more 
minors and the determinant :— 
dd TIT mp PP Port [Ve dd 1) ) saa TH n® dP 
wy b Pgh TO gt + aoa ast it gt te” .2? + a’ oa§ 
Q, @ +0 a +e 0 +d a, a! +b 0 + x? + 8 
da hn TIP pnd Lid i 3 dd PES vn JEP XS etre 
ry a Be + Ca? + Ba, a + Oat + e/a? + a!" 428, 
/ 
a8 
so that the original determinant is resolved into the sum of six minors of the 
form :— 
A, +B,7+C,2?+D,2°, A. + Br + 0,2? + Dr" 
A, + Bx + C2? + Dov’, A, + Bux + Ca? + Dia, 
3. Account of the Levelling Operations of the Great Trigonometrical Survey 
of India. By Major A. W. Bairp, R.E., F.R.S.—See Section E, p. 1123. 
4, A Theorem relating to the Time-moduli of Dissipative Systems. 
By Lord RayuericH, D.C.L., LL.D., F.R.S. 
In the proceedings of the Mathematical Society for June 1873, it is shown that 
the times of vibration of a conservative system fulfil a stationary condition, so that 
the time of vibration in any normal mode would remain unaltered, even though 
the system, by the application of suitable constraints, be made to vibrate in a mode 
slightly different. It is pretty evident that a similar theorem must obtain for the 
time-moduli of the normal modes of a dissipative system, but a formal statement 
may not be useless. 
The class of systems referred to is that of which the mechanical properties depend 
upon ¢wo functions, one being the dissipation function F, and the other either the 
‘kinetic energy T, or the potential energy V. As examples of the first case may be 
mentioned, the subsidence of the small motion of a viscous fluid contained in a fixed 
envelope, and of free electric currents in a conductor. On the other hand, in the dis- 
tribution of heat in a thermal conductor, or of electricity in a cable, the undissipated 
energy is usually regarded as potential. The argument is almost exactly the same 
whichever case be contemplated ; to fix ideas we will take the former. 
By suitable transformation the two quadratic functions T and F may be reduced 
to sums of squares of co-ordinates, and these co-ordinates are consequently called 
normal. Thus:— 
T= cr $7,+3 [2] 7+... 
F= 3(1) $’7,+4(2)o7+... 
in which all the coefficients [1]... (1) . . . are positive. 
The normal modes are those represented by the separate variation of the co- 
ordinates, and the corresponding differential equations are of the form :— 
[s]ps + (s) ds =0, 
d. = Pe-» 
p=(s)/[8] 
If r, be the time-modulus, the time in whieh the motion is diminished in the 
ratio of e : 1, t,=p7}. 
Suppose now that by suitable constraints an arbitrary type of motion is imposed 
upon the system, so that ¢,=A,6, d,=A,6, ... where A,, A, &c. are given 
(real) coefficients. Then 
T= {[1JA+3[2]A,?+ ... 6 
F={i()A,?+3(2)A.7+ ... A; 
~whence 
-where 
