140 Prof. A. M c Aulay on Spontaneous Generation 



It is easy to see that (29) gives the most general form of w, 

 consisting of terms in s 12 , s , and s'", which is a true minimum 

 for e=f=g=0. We can make the state corresponding to 

 the south valley truly stable instead of unstable by adding 

 to the right of (29), ^(S^ 2 ) 3 , or a higher power of 2^ 2 , k 

 being a positive constant as small as we please (so as not to 

 alter the values of w for small values of e,f, g). 



Our hypothesis requires that there should be at least one 

 set of small values for e, f, g besides e=f=g = 0, giving a 

 stationary value of w. For such small values we may neglect 

 orders above the third, and in the third we may replace 

 e, f, g by e l9 / l5 g v Do this and put 



—«i 4/1+^1= + 2 JPi ^1-/1+^1= + % ^i+/i-i/i=+2r,l 

 + e 1 = q^-r 9 + f { = r+p, + g 1= p + q. J 



Thus we get 



w/A=-6^ + &2(?-r) 2 + c(2p) 2 . . . (32) 

 For the stationary values 



Using these in the form 



iv p =-0, lV p —W q =Q, W p —>lV r =:0, 



it can be easily showm that a stationary value necessarily 

 requires two of p. q, r to be equal. Taking these as q, r, 

 we find as the only cases besides p = q = r = the following 

 for stationary values of iv ; either 



2j = q = r = c, (33) 



or else 



q=zr=—b, 2 J = a > (34) 



where 



a=b(b + 2c)/(2b + c), .... (35) 



so that a is positive. 



c might be finite or even much larger than unity, and still 

 if b were small there would be a stationary value for a small 

 strain besides the minimum value at j9 = ^ = ? 3 = 0. In this 

 paper, however, we will suppose that c is small and b very 

 much smaller. 



The surface (24) is obtained by putting a—p, y = q=r, 

 £ = iu/A. The quadrants mentioned above are determined 

 by y=- —b and x— — b. These two straight lines pass through 

 the extreme south and west points of the lake. [If b<c 

 similar remarks apply to the contour line passing through 



