( 470 ) 
my’ + 6b,y* + Shy 
3b,y° + 6b,y + m 
— (20) 
reduces (14) to (19). 
5. To determine in this case the conditions, we differentiate the 
b 
° expressed in 4 and u by (15). This gives 
il J ] 5 ‘ 5 
four values ET 
GOB) = (BB apty—BB,,)2,'—(B,p1, + GBy)Ag! + batt! HBA | 
+ (DA, + 6b3,)u9 —b,A.u,'—36,a,u,' 
64(b,by') = (3b ,4, —3b,u,)A,' — bq) + (b,42—3b9)A,' + 36,2.) + 
+ b,2,u2' + (8624,—b,49)u,' —36,2,u,. 
64(bb9'J=—3b eds + (Bbout,—b,U, )ag + 6, My4,' H(3D,— Bbz), + if 
+ (6,2, —3b A, ta —b Agu,’ + (8b242—3b,a,) yo 
64(b,by')=— 3b, ,A,,—b uy hg + (b fa 6bou)d, +(3b,—6bou)d, + 
+ Did,’ —(b Ag+ 6b94,)u,' + (6624,—3b,2,)u,. 
(-1) 
Combining each of these equations with the seven former equations 
(15) and eliminating the quantities 2, 25 2,'à, u u, #,… we obtain 
a es ee 0 0 0 
|% Wz i oD. 0 A 0 
| dB Po derd hoes 0 
| Ce, ig Oh ea eee wig 
a O Uy Bi Uy —4, Ah Ag | 
a, © B dE 0 —A, —A, 
aq 0 Dk: 0 0 —aA, 
ellis We Wa en CG: C, 
where the last row is formed by the coefficients of each of the 
four equations (21). Hence for the first of these 
Ge == 6(b,52). C, == BboUt2 = 36 Uy, etc. 
If we reduce this determinant in the same way as before, the last row 
becomes in the first place 
CHC, ACC, CdC, 
6 lo, 4 6 
Cor Cyr Car Csr Cos 
and secondly 
