OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 529 
where 7), Pz... +. P, have the values given by the equations (III.), ana ¢,, ¢... are 
treated as constants in the differentiation. 
Since Q = 0, we may use instead of the operator A, another, V, such that 
de, 0 dé, 0 
V = Alp Oe, + ae Oey + Ondo 

for AO + VO = 02 _ 
By comparing VO = 0 with (VI.), we see that all the determinants of the matrix 



02 OF, oF, OF, 
Oa) sandeue aoc rea ince 
0 OF, 
i aoe 
02 ak, 
OC, 4 Oe, peas 
vanish. 
§ 8. For the third singular solution we take the further integral equation A°Q = 0 
or V*Q = 0, which forms are equivalent, since already Q=0,AQ=0. The argu- 
ment is the same as in §6, and may be carried on till the last singular solution is 
reached. 
The equations O=0=VO=V70=...=V'!0, which yield the 7 singular 
solution, show that the equations F, =0= F,=F,=...=F,, when solved for 
Cj, Co, .- + €,, have » + 1 coincident solutions. 
§9. The equations in the other form, viz, Q=0,AQ=0... A”’02 =0, show 
that the system 
1D ei) Se) De iy, Mies aK) 
is satisfied by r + 1 consecutive sets of values of , y),... Yn 
For suppose E to be the eliminant of F,, F,... with respect to ¢, .... ¢,, we 
have then an identity 
E = A,F,+4,F,+...+A,F, + Bo. 
Differentiate partially as to c,, c,... ¢, in turn, and we have 
19) y oF. oB Y oA 
pees en gee east > ye — 9 
B 26, - =A, 0, o5, ZF 2c, (Gis 1) 
Eliminating A,,... A, on the left-hand side, we find 
ay Broa al) e CEN and) ee © aan Yep Spa Dy) 
2 Gil ren St) ay Be a) OCae cs) ze E, Gi(@p calc Gn) 
MDCCCXCV.—A. 3 We = 
