564 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where we are at liberty to suppose 
dp -^ — qcfoq, • • • • j d p n — i n dx n , dcr i i* dy^ • • • • » d<r n 
Our typical quadruple integral has a boundary which in a definite sense 
may be said to be of three dimensions or trebly infinite in third order 
infinitesimals. In our mode of picturing the integration this three- 
dimensional boundary consists of eight parts corresponding to the upper 
and lower limits of each of the four parameters u, v, w, z. Now a triple 
integral over this complete boundary can always be expressed as a 
quadruple integral extending between the same limits as our original 
quadruple integral. For if we sum the triple integral over the boundary 
of each of the quadruply infinite four-dimensional elements we shall get 
by cancellings the integral for the original boundary only. 
If the element of the triple integral is (f>S. 6 dp 1 dp 2 dp 3 the element 
(dp v dp 2 , dp 3 . , dp 4 ) contributes four pairs of elements whose sum can be 
shown to be 
<fi b ($sdp 2 '7 p^dpjSdp-^^/ b — Q.^dp^dp^dp^dpc^h 
+ %d Pl dp 2 dpfidp^j b - S. 6 dp 1 dp 2 dp.fidp 4 \^ b ) 
= i>b^ 9 (\Vb^dp 1 dp 2 dp s dp 4 ) 
by (14) § 5. Thus 
pc- 1 ) r pc) r 
) jWc-^-j l<f>A-i( \Vrdp?) . . ... ( 17 > 
the (c — l)-ple integral on the left extending over the complete boundary of 
the c-ple integral on the right. This is, of course, the generalisation of the 
quaternion line-surface and surface- volume integrals, and includes both. 
Putting <p = l in (17) we get J (c_1, J(7pjLi 1) = 0 for such a complete 
boundary. This is a fact, but cannot be regarded as a proof, as the fact 
is required for the quadruple element in the steps indicated by the words 
“ can be shown to be.” 
Since {da/dp} c converts dp ( c c) to da [ c) we may extend the da I dp notation 
according to the following formula, 
d Xl = i d XL l =s( )|A (C) o- l<: 
\d P L W1, ■ 
(To-W 
1 = S( )\o-v a (£) = pV 1 = pV? 
d‘p? \ d'p I c U1 0 • 1 „A? M 
ls( )+f r + < y>+ 
[dp ) dp dp 0 
d<T { n l) 
dM 
. (18). 
. (19) 
. (20) 
These serve to imply how intimately the fictorlinity replacement is 
connected with general strain (a an arbitrary function of p) in Euclidean 
space of n dimensions. 
