DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 



41 



or 2 1 in all. There are certain relations among the differential equations ; and 

 further, four integrable combinations of the new system are known to exist, viz., 

 the initial equation F = and the three equations derived from it. What is 

 wanted is, if existing, a new integrable combination. 



21. Thus far the analysis applies whether the characteristic invariant is or is not 

 reducible to two linear equations. Suppose now that, as in 15, it vanishes, in 

 virtue of one or other of the two equations 



= 



Ap 



0)q- (iG 



and consider these in turn, in the same manner as before. 



Then, by combining the equations of identity with the equations particular to 



F = 0, we have 



(XX) + AS + (HI + &) 8& + (AG + 6ty) Sa t == ] 

 (XY) + A8& + (ill + 9) 3 7u + (i G + 0<f>) Sft == ' 

 (XZ) + AS ai + (ill + 6) S/3, + (r^G + <ty) 8 3 == 

 (YY) + AS yu + (J H + 0) 8S + (J-G + 0</>) 3 7] = 

 (YZ) + A8& + (4H + 0} S 7l + (G + 6$) S& = \ ; 



J 



(ZZ) -f A8 3 + (^H + 0) S& + (^G + 04.) Say = 

 as a modified form of the equations for 



Ap + (Ul - 0) < L - (1G - 6$) = ; 



and in this form 

 Let now 



s d . %H-0 d 



8 = <>., + A dy 



E (a , . . ., 8 , . . ., h, I, m, n, v, x, y, ;) = 

 be an integrable combination, so that we must have 



BE da SE 



** J ~ J t& W fl I -A I 



5 V" i 2 3~ , ~r 



r\*4 ft & f> ft ft ) ' 



( OCy tCrf- (-.[t/ tltt- 



, ,, . v L 



+ v (* + np) + p - + p ^, = 0, 

 do cs 



, OPJ 



S "^r~ 



Multiply the first by A, the second by ^H 0, and add ; then using the equation 

 connecting p and q only, we find 



A3 + A + ( 4 H - 0) + (4G - W - 0, 



where 



VOL. CXOJ. A. 



