538 
FUNCTION. 
[787 
h, k; we must therefore have Ah + Bk, Gh + Bk = \h — /xk, fxh + \k respectively, that is 
A, B, C, D — X, — ¡x, /x, X respectively, and the equation for tan 6' thus becomes 
tan 6' = 
/x + \ tan 6 
hence writing - = tan a, where a is a function of x, y, but inde- 
A, 
\ — /x tan 6 
pendent of h, k, we have tan 6' = ^ an a ^ an that is 0' = a + 6 ; or for the given 
1 1 — tan a tan 6 ° 
points (x, y), (x\ y'), the path of P being at any inclination whatever 6 to the axis 
of x, the path of P' is at the inclination 6 + constant angle a to the axis of x ; 
also (X/i — fxk) 2 + (fxli + \k) 2 = (X 2 + fx 2 ) (h 2 + k 2 ), i.e., the lengths of the paths are in a 
constant ratio. 
The condition may be written S (x + iy) — (X + i/x) (Sx + ièy) ; or what 
thing 
d + ^ i %) Sx +d +i %)= (x +^ (&+iiÿ) - 
that is, 
ë + d =(x+ * il) ’ 
consequently 
dx' . dy . (dx' . dy'\ 
dï +t iï = t lxc + l <ü) ; 
that is, 
II 
^ № 
S3 
1 
II 
^ l^s 
is the same 
as the analytical conditions 
obviously imply 
in order that x' + iy' may be a function of x + iy. 
d-x' d 2 x d 2 y dry' = _ 
dx 2 dy 2 * dx 2 dy 2 ’ 
They 
and if x' be a function of x, y, satisfying the first of these conditions, then 
is a complete differential, and 
dx' 
dy 
dx + 
dx' 
dx 
dy 
18. We have, in what just precedes, the ordinary behaviour of a function (p(x + iy) 
in the neighbourhood of the value x + iy of the argument or point x + iy; or say 
the behaviour in regard to a point x + iy such that the function is in the neighbour 
hood of this point a continuous function of x + iy (or that the point is not a point 
of discontinuity): the correlative definition of continuity will be that the function 
<f> (x + iy), assumed to have at the given point x + iy a single finite value, is continuous 
in the neighbourhood of this point, when the point x + iy describing continuously a 
straight infinitesimal element h+ik, the point cf>(x + iy) describes continuously a straight 
infinitesimal element (X + i/x) {h + ik); or what is really the same thing, when the 
function (x + iy) has at the point x + iy a differential coefficient.