144 Prof. A. M c Aulay on Spontaneous Generation 



Hence the quantities m\ e, m", m" f , Se%6 are found to be 

 adequate to express s f , s n , s' 1 ', always excepting the fourth 

 order part of s"', which is left unmodified below. 



(43) and (44) below are given for reference. (45), (40) 

 (47) contain the modifications we seek. 



($ Vv ) 2 =™' 2 



SViV 3 S7 ?1 77 2 = m' 2 -2m"-e'4, . . . (43) 



(SV^) 3 =-m' 3 



S V^S Vi V 2 ^ViV^ = ~ m'(m'* ~ 2m-' - * 2 ) 



SV^SVi^SV s ^i= - '»'K 2 -2m // ) 

 SViV*SV«%Sihi/8= -m' 3 + m'(3m"+e 2 ) -3m'" -Sex* 

 SVi^2SV 2 %SV3^i=-w' 3 +3mW / -37^" / 



/=m' + l(m' 2 -2m , '-e 2 ), .... (45) 



*"= l^-im^-^-m'", . . (46) 



s"< = ( - 1.W 3 + 2m W 4- We 2 -4m'" - Se % e) 



-iSViV 2 V 3 V 4 Sw7 3 S97 2% . . . (47) 



The third order terms of /' are equal to —4 times the 

 discriminant of %-^\m that is of — ^VVi( )vi* [The dis- 

 criminant of any linity (j> means -J-Sfifg^S^fj^gj^fs]. For 

 the discriminant of % + # is known to be 



m"' + am" +aPm' +a?, 



so that the discriminant of %'—\m is 



m'"-lm'm"-f |-m' 3 . 



Also the discriminant of </> is known to exceed the discri- 

 minant of </> by ^S\<£X, where A is the curl V£</>? of <£. 

 Putting ^> = ^— ]m', X becomes 6. There is probably here 

 some easy method involved of proving (47), but I do not 

 see what it is. [What is required is an easy proof that 

 -2^ViV2V3?72S*7i?73— 1jSViV 2 V 3 S?7i?72??3 is — 4 times the 

 discriminant of % — -JW.] 



[Ptm? Mathematical Note. — The whole theory of Unities 

 of the nth order, and especially such work as the present, 

 may be based simply and very instructively, on the study of 

 the scalar n-linear symmetrical function which in the case 

 of n — 3 is given by 



