OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE 
340 
17. A special case of these six equations is discussed'"' by Professor ^orawski in 
his memoir already quoted, viz., that in which there occurs a single function 6 with 
its derivatives up to the second order inclusive, and there are no derivatives ot 
E, F, G of order higher than the first; and he obtains three independent solutions. 
These are 
(I — 2\'^oo + (E|,i — 
h = - 
r = — 
) 
I 
^Ol'’ + 
Manifestly, a, h, c are independent solutions of the e(|nations in the jjresent case ; 
also, other three independent solutions are given by 
+ (Eoi — 2F]|j) p — 
1/ = - G,„p - 
(• = + (G||| 
Ei,-,cr 
Koi- :■ 
All these six solutions are inde})endent of 6: tq, vq, ; r,. r.,, r.,, r,. 
The JAtTiBi-PoissoiM' conditions of coexistence of the six equations are satisfied 
either identically or in \irtue of the eight equations (lllj) to (Hlg), which are 
definitely satisfied; so that, taking account of the variables that occur in the 
derivatives of /, the set of six equations is a conqilete system. The number of these 
variables is 
(J, from the first derlvati\'es of E, F, G, 
+ 6, . . . second . . . . (fj, \jj, 
+ 1, being 0, 
+ 8, lieing ?i,, v^, rq, Vo, v^, 
= 21 in all; hence the total number of algebraically Independent solutions of the 
complete system of six equations is 15. Of these, we already possess six in 
a, h, c, a', b', c', so that other nine are required. 
The form of the equations suggests that there will be four solutions of the type 
Un + aP„ + bQii + + kS„ , 
four of the tvpe 
+ «'Pb +//Q'. + e'Rh + Sh, 
and one of the type 
^ + T,,; 
which, when obtained, will be the necessary nine. 
18. One mode of obtaining these solutions is as follows :—We use the values of 
a, h, c, a', h', c' to eliminate from /’ the second derivatives of and of xfj ; the effect is 
* Loc. cit., § 2(5. 
