166 Proceedings of the Royal Society of Edinburgh. [Sess. 
proves preferable, a set of twenty-one integrals, subject to a single relation) 
can be taken as follows : — 
<^44 ^11 ~ ^ 1 
/^4^ — 2t?? 24 -f C?22 ~ > 
^44 “ ^^34 “t ~ y ’ 
^33 ~ ^/23 *^22 ~ ^ ’ 
^33 ~ + ^11 “ > 
^22 ^^12 ~ ^ } 
«23-^63~.'7i2+/ii = '<, 
^24 “ ^^4 ~ ^2 + ~ X ’ 
^34 “ ^^14 “ ^3 + ~ ’ 
^13 ~’/l2 ^'^23 ^22 “ ^ » 
^14 ~ ^12 ~ ^^24 ^22 ~ P ’ 
^34 ~.^24 “ ’^^^23 ^''22 “ ^ ’ 
'"12 ~/l3 ~ 928 ^^33 ~ ’ 
'"14 “ ^13 “ ^^34 ^33 ~ ^ » 
^24 ~ ^23 ~/34 ^^^33 ~ ’ 
'^12 “ ^24 ~ ^^14 ^'44 ~ ^ ’ 
'^hs “ ^34 “ '^14 .'^44 “ ’ 
'^23 ~ ^^^34 ~ ^24 "^"/44 ~ ^ ’ 
/l4 “ .^24 ~ ^^^13 "t ^23 “ ^ > 
^^34 ~/l4 “ ^23 ^12 ~ ^ ’ 
924 ~ ^^34 ^12 ^^^13 ~ ^ ’ 
with the single relation 
V + i/' 4- w — 0 . ' 
It should be noted, in passing, that the twenty integrals (except for a 
factor i) constitute the terms which involve derivatives of the second 
order occurring in the twenty independent Riemann-Christoffel symbols. 
The Second Sub-Geoup: Forty Equations. 
19. Next, we proceed to the second sub-group of equations. These 
arise in a manner similar to the former sub-group, viz. by the considera- 
tion of the coefficients of the second derivatives of i, rj, 6. In all, there 
are forty equations in this sub-group. 
When the equations are formed, it appears that they involve (among 
other quantities) second derivatives of the magnitudes a, h, c, d, f, g, h, 
m, n. The only admissible combinations of these derivatives are now 
the preceding twenty integrals in the set just obtained. Consequently, 
the forty equations must be transformed so as to substitute these com- 
binations as the variables in place of the scattered occurrences of the 
