*99 
the calculus of junctions. 
yet they do vary in a certain sense, because y is differently 
contained under each functional characteristic ; the method 
of treating these kind of equations will be explained hereafter. 
Problem IX. 
Given the functional equation 
V'X*,y) = o 
This signifies that the second simultaneous function is equal 
to zero. It is evident that oc — y ory — sc will be a particular 
solution, for if ^ (a?,y) = sc — y we have 
V’ X x > y) = ^ Ot' 0 *^ ) y ) ) = O — y) — (°° —y) — 0 
By observing the process just gone through, it appears that 
it would equally succeed if for sc we put/ (oc) and for y we 
put/ (y) for if -fy (sc, y) —foe — fy, we have 
X x >y) — (Jx—jy) — Cff —fy') - 0 
This solution is considerably more general than the former, 
yet is by no means the complete solution, a more general one 
may be obtained thus : we found one particular solution to be 
(a?,y) = sc y, now if we multiply the right side of this 
equation by an arbitrary function of a? andy the condition will 
still be fulfilled ; for if $ (T,y) = ( oc — y) <p (a?,y) we shall find 
q/ 3 (a?,y)== |a?-y<p(a?,y) — a? — y <p(a?,y) j- x 
<P { JV <P O^dO } = 0 
provided ja?— y <p (a?,y), — y (a?,y) j does not contain in , 
its denominator any factor which vanishes. 
