142 Prof. A. M c Aulay on Spontaneous Generation 



From this we obtain that T'T or (l-fijr) 2 depends only on 

 the pure strain, w may therefore be regarded as a function 

 of T ; T or of v because 



y=i(T'T-l) (37) 



It will be noticed that the roots o£ yfr are e, f, g ; the roots 

 of v are e + %e 2 ,f+ \f* 5 g + ig*; and the roots 'of T'T are 

 E 2 , F 2 , G 2 . 



We may work with 



/>',T,TT,E r F,G, 



or instead we may work with 



V,X^e,f,g. 



In this paper the latter course is almost exclusively followed, 

 but we do not consider ourselves confined to it. 



We first find the simplest forms of s', s", s"' in terms of 77 

 and its space derivatives. We have 



mi =m' + m' / W"=-SV^~iSVViV 2 Vw ?2 



+\8ViVf7&viv*o*> • • (38) 



5 '=:-S?i;f=~S^=-SV^+iSViV 2 S W2 . . (39) 



From these we form by the same process as in (26), (27), 

 (28), s' r and s'" in succession thus 



s" =s'-m 1 =iSViV< i ViV2~l$ViV< i VSviV2V3 • (40) 



for s'" was obtained by subtracting from s" that definite 

 quadratic function of e + \e 2 , &c, which cancelled the second 

 order terms. The only quadratic function of v which can 

 effect this cancelling is that one which is of the same form 

 in v as the quadratic terms of s f/ , namely 



are in %. 



We now show that 



Sfl£2U?il^2 = S£if 2 U?il£2> 



by putting i> = u-f-- 2 -Ve( ), and showing first that the linear 

 terms in e, and secondly that the quadratic terms in e vanish. 

 As to the linear terms we have (Th. l)j 



