172-175] 
As in § 169, let ® 1} © 2 , ... be a configuration of equilibrium corresponding 
to this given value /a of the parameter, so that in this configuration 
—= =0 (174-2). 
B0 x d0 9 d0 t K ' 
In any adjacent configuration © x + 80,, © 2 + 80 2 , ... fi + 8 fi, the value of W 
may be expressed in the form 
dW d*W 
8SSS+- 
3 W d 2 W d 2 W 
+ w + (8/i) (m s^M, + - 
(174-3) 
and the condition that this new configuration shall be one of equilibrium is> 
from equations (174"2), 
32 w 3 2 w 
êôl W + 802 ddM + 
32 w 3 2 Tf 
+ *Æ + *‘ mr° -< im > 
and similar equations. Writing W 12 for d 2 W/d0,d0 2 and so on, the solution of 
these equations is 
Ml (174-51 
w 12 , w, 3 , ... w lfL _l ( 
W&, W 2 3, ... w 2 
where A is the Hessian of W with respect to the variables 0 1 ,0 2 ,... 0 n , given by 
w u , w, 2 , ... w, n 
W 21> Wn, ... W 2n 
.(174-6). 
The values of the ratios 80,:80 2 : ... : Bfj, determine the ratios of the small 
changes in ©j, © 2 , ... /j, as we pass along a linear series. At points such as 
Q in figs. 20 (i) and 20 (iii) one or more of these ratios must become indeter 
minate, so that we must have (say) Bfju/B0 s = 0. At a point such as Q in 
fig. 20 (ii) we must have (say) B/j./B0 r = 0. Thus the three points Q in fig. 19 
are all determined by the single condition 
A = 0 (174-7). 
175. If fi is kept constant, the change of potential energy corresponding 
to changes 80,, B0 2> ... in the values of © 1} © 2 ,..., will be given by 
8 W = ^(80,f W„ + (80,) (B0 2 ) W , 2 + (175-1) 
in which no terms of degree beyond the second need be written down, since 
80,, 80 o, ... are supposed small. 
Let the co-ordinates 80,,80 2 , ... in this quadratic expression for 8TT be 
changed by a linear transformation to new co-ordinates <f),, <j> 2 , ..., such that 
8 W becomes a sum of squares, say 
8 W = \(b,<t>, 2 + b 2 $ 2 2 + ... + &„</>„ 2 ) (175-2), 
the modulus of transformation being \. 
BB mm 
♦V ÎJi’J 
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