130 Prof. A. M c Aulay on Spontaneous Generation 



negligible compared with rj. %, T, and v all have the same 

 rotation vector, namely, ^VV*?- We shall write 



e = VV% (6) 



and call e the curl. For finite strains the cnrl thus defined 

 is not correctly described as the double rotation. 



We shall speak of % and e as of the first order because 

 they are of the first degree in 77 ; similarly T is of the zeroth 

 and first orders, v of the first and second orders. 



The ratio ch' jcU of the strained element ds 1 of volume to 

 the standard value ch will be denoted by m ; it might be 

 called the total dilatation or total expansion, m — 1 being 

 called the dilatation or expansion. The three parts of m of 

 first, second, and third orders will be denoted by m r , m" , m' ff . 

 Thus 



m=d$'/d$=l + m! + m" + »»' ;/ \ / 7 \ 



=iS7iV 2 V s S Pl 'p 8 ' / >3 J * • * U 



m'= -SV17, m"= -iSVViV 2 Vi?i%, m"' = JSViVsVsSwMs 



(8) 



m' will be called the divergence and — m' the convergence. 

 [Thus in this paper the curl, the convergence and the 

 divergence stand for the curl of rj, the convergence of rj and 

 the divergence of »?.] 



The density in the standard and strained states will be 

 denoted by ?i, it, so that 



nck = n'd<;', n' = m~ l n (9) 



A vector element of surface in the standard and strained 

 states will be denoted by d%, d%', so that 



d P ' = Tdp, dX^mT-hlX, d$'=mds. . . (10) 



The potential energy of the element, ^?, will be denoted 

 by wd<s ; also if convenient by w'ds 1 . The characteristic of 

 a fluid is that w is a function of m only ; of a solid that w is 

 a function of v , that is of T'T, which is not a mere function 

 of m. 



If 2 is a scalar function of v. the differential operator 9 

 is defined by 



dz^Sdvtfrzi;. (ii) 



Thus 9 is a symbolic self-conjugate Unity capable of explicit 



