176 Proceedings of the Koyal Society of Edinburgh. [Sess. 
The first minors of A are taken in the form 
Ajj = bed + 2 fmn — hrd‘ — ail'd — df^, 
A22 = eda + 2 gnl - cd - ard - dg-^ 
A33 = dab + 2 hlm — and - bd — dhd, 
A^^ = abc + 2 fgh - af^ - bg- - c/d, 
Ai 2 = Aoi = dm + d/g + hn^ - cd/i. - gm 7 i -fnl, 
Aj3 = A3j = hnl + d 7 im + g 7 id — hdg —fhn — hmn , 
A03 = A32 = a 77 i 7 i + dgli + fd - adf - /ml - glm , 
Aj^ = A^j = bgn + c/mi + ip - hcl - /ifn - gf 7 ii , 
A04 === A42 = dll + afn + 7 mp — ca 7 ii -fgl - hgn , 
= A^3 = af 77 i + hgl + 71/d - abn - ghm - f/il . 
23. Noting therefore (i) the forms of the coefficients in the sub-group 
of forty equations, (ii) the fact that there is no integral which involves 
only derivatives of the first order, and (iii) the fact that the number of 
independent integrals required is the same as the number of Eiemann- 
Christoffel symbols, we are led to surmise that the integrals can be 
associated with the s^/mbols, member by member. The surmise proves to 
be justified in fact. I have found twenty-one integrals in the following 
forms, the symbols (ij, i'j') in the first line being the actual Riemann- 
Christotfel symbols * : — 
A = (41, 14) 
s t 
sf 
11 
44 
14 
14” 
s 
t 
s 
t _ 
B = (42, 24) 
C = (43, 34) 
( 
St I 
' 22 " 
s 
■44 
t 
p][ 
24 
t 
/ ’ 
33” 
44” 
”34 
34 
s 
t 
t 
/’ 
D-(32, 23) 
.is+i22A.,{p][f]-pjP=>] 
* The reason for the apparently arbitrary notation A, . . ., V for the integrals will 
appear later (§ 26). 
f In the double summation for s, ^ = 1, 2, 3, 4, all the combinations are to be taken. 
Thus we must retain 
while Aj 2 = Aoi ; and so for other combinations in A and in the succeeding integrals. 
