L178 
MR. R. F. GWYTHER OR THE DIEFERERTIAL 
Write 
x = X, ^ t y = Y eX, rj = 7) + e^, 
where, e being infinitesimal, we shall only retain its first power. 
Then, by hypothesis, 
/(^ -X, V -y, Vi, Vi,. . .) -X, rj-Y, Yi, Yg,. . .). 
Also expanding by Taylor’s theorem 
/(^ -X, rj — y, y^, y^,. . .) 
= f{^-X,y-Y, Y„ Y^,. 
..) + €{(f-X) 
3(,-Y) 
Therefore 
(4)- 
Hence we come .to the conclusion we arrived at before, that 
7] — Y and Y^ only enter in the form rj — Y — Y-^ — X), or tt — P. 
We shall now reintroduce tt — p, and write the functions 
— X, TT —p, yz, 2 / 3 , . . .), &c. 
The method we have used here we shall use in the other cases. We evaluate the 
multiplying factor to obtain one expression for — x, tt — p, .. .), and obtain a 
second by expanding the function in accordance with the transformation considered. 
3. Having now introduced tt — p, we shall treat the other infinitesimal trans¬ 
formations by a more general method. 
Consider the homographic transformation 
X 
y 
Y 
1 
X + BjY 
AX -f BY + 1 
where B^, A and B are indefinitely small quantities whose squares and products will 
be neglected. 
Thus 
X = 1 + B^Yi + B (Y - XY,) 
= 1 -f AX + BY 
V ^ A^ -j- Bi^ 
