120 
5. The theorem however does not cover the whole ground, 
for if V be a variable function U + Y=:0 may give a solu- 
tion. This is an easy deduction from (p. 116 of) a paper of 
mine cited in Art. 7 {infra). 
6. In my paper “On Ternary Differential Equations” 
{ante, vol. xvi., pp. 66 — 68), in connection with which that 
of Mr. Eawson, bearing the like title {ib., pp. 114 — 118), 
should be read, ~ and ^ must be interpreted as follows, viz. : 
dx dx ^ dz dy dy dz 
and this being done, (3) is an equation which becomes an 
identity if, in the sinister, we replace p and q by M and N 
respectively, or operate vice versa on the dexter. In what 
follows I recur to my paper. 
7. At p. 115 of a memoir printed in the Proceedings of 
the London Mathematical Society (vol X., pp. 105— -1 20) I 
have in effect shown that 
x dM /lv/r TT 
( 7 )- 
This numbering (7) is, and is intended to be, consecutive 
to that of my paper on ternaries, and does not occur in the 
memoir. 
8. Subtracting (7) from (3) we get 
“-^ =0 (8)> 
and we get this same (8) by transposing the sinister of (7) 
to its dexter. 
9. Comparing (3), (7), and (8), we see that IT vanishes 
when 0 is eliminated by means of M =p or N = q. 
10. Let z-y-x=u and M = 1 + u m { 1 + au n - bu r ), N = 1 4- xu m ; 
then ^ 4- N~ = mxu m ~\ M - 1) + xu^^nau 71-1 - rbu r ~ l ), 
^ + = + ( M ~ 
U = — u m 4- xv? m {nau n ~ 1 — rbu r ~ 1 ). 
