the calculus of functions. 
223 
and putting <p* x for x and (p y for y we have 
F { <p W, v'y,Tf{x>y)> ?'/'■ 2 fay), f/ 2 ’ 1 &c.}=o 
Some particular value of f must now be assumed, and the 
equation treated as one of the first order relative to <p . 
The form to be assigned to/ is of some consequence; it 
ought to be a particular solution of the original equation : for 
if we assign to it any other form, this adds a limitation to 
the original equation which may or may not agree with it, 
a particular solution should therefore always be employed. 
This remark is applicable to several Problems in my former 
Paper, and with this restriction, their solutions will remain 
correct 
Problem XXVII. 
To transform the equation 
F | x,y, ^ (x,y), -ty 2 > 1 ( ax , fiy), 2 (ax, fi y ) , &c. j = 0 
into the form of the equation of the preceding Problem. 
Assume -ty (x,y) =s (p r /(( px, <py), then the equation becomes 
F {x,y } ff(cpx,(py),(p f 2>1 [(pax, $fiy),q>f l,z <t>fty)A c - } =0 
find for < p by Prob. VII. Part I. such a value that it shall not 
change when any of the following quantities are substituted 
for x. 
aX 
fix 
UX 
fix 
X 
1 
UX 
fix 
V. 
% 
&c. 
&c. 
let this value be A, then the equation becomes 
F {x,y, A' f {Ax, Ay,) {Ax, Ay), &c.} = o 
