588 
ON THE CONDITION FOR THE EXISTENCE OF A 
[625 
Combining herewith the equations obtained by differentiation of a 2 + /3 2 + <y- = 1, viz. 
da. d/3 dry 
“- + /3 ® + VTÍ= 0 . 
dx 
da 
dx 
dry 
a d/3 „ _ 
! ‘d¡j + íi dy + ' f d'y = i) ’ 
cfo dz 
dz 
and subtracting the corresponding equations, we obtain three equations which may be 
written 
a . P . _d/3 dy dy da. da d/3 
' dz dy dx dz ' dy dx ’ 
or, what is the same thing, 
■ i 
d/3 dy dy da da d/3 _ . 
dz~"dy’ dx~dz’ dAj~~dx~ k(l ' ky ’ 
and, multiplying by a, /3, y, and adding, 
k — a 
(<№ _ <h\ o (dy _ da\ 
\dz dy) dx dz) 
We thus see that, if the function on the right-hand vanish, then k = 0, and conse 
quently also 
d/3 dy dy da da d/3 , 
dz dy ’ dx dz ’ dy dx ° aC 1 — ’ 
viz. if the equation adx + /3dy + ydz = 0 be integrable, then adx + /3dy + ydz is an 
exact differential; which is the theorem in question. 
But it is interesting to obtain the first mentioned set of differential equations 
from the analytical equations of a congruency, viz. these are x=mz+p, y=nz + q, 
where m, n, p, q are functions of two arbitrary parameters, or, what is the same 
thing, p, q are given functions of m, n; and therefore, from the three equations, 
m, n are given functions of x, y, z. And it is also interesting to express in terms 
of these quantities m, n, considered as functions of x, y, z, the condition for the 
existence of the set of surfaces. 
We have 
«, ft 7 = 
in 
R’ 
n 1 
R’ R’ 
where R = Vd + m- + n 2 ; 
and thence without difficulty 
d n d d\ 1 
«- + ^7T.+ 7 S )« = S - i 
dx ^ dy 1 dzj 
oW dm dm dm 
mn 
dn dn dri 
m j—b u j—I—j— 
dx dy dz 
— mn 
)) 
+ (1 + m 2 ) ^ 
»