80 A Constructive theory of Partitions, arranged in [1 
i] 
contour. As this contour expands uniformly in all directions through 0, the 
line 'AA' remains parallel to itself. Since 'AOA' is an elementary triangle 
so also must the similar triangles 'AAA', A'AB', 'AA'B be all elementary, 
consequently A will be the first new node intervening between 'A, A' 
brought into the enlarged aggregate as ’AA' moves continuously parallel 
to itself, and 'AO A, AO A' will be elementary triangles; it may be noticed in 
order to bring this method into relation with that indicated by Mr Glaisher, 
that the coordinates of this new node A are the sums of the coordinates 
of its neighbours 'A, A'. If the contour were not supposed to be simple, 
this condition could not be drawn; for if there were a hole round the middle 
point of 'AA' the node A would be missing in the enlarged aggregate, and if 
the first node to intervene as the contour went on enlarging be called (A), 
'AO (A) or (A) OA' or each of them would be a multiple of the elementary 
triangle, so that the constancy of the value of the successive determinants 
would no longer hold. In like manner it will be seen that on the same 
supposition as above made, if in consequence of the contour contracting about 
0 as the centre of similitude, two points 'A, A' which originally are non 
contiguous, at any moment become contiguous, at the moment previous 
to this taking place A (and no other point) must have intervened, and after 
A has disappeared from the reduced aggregate, no other point can make its 
appearance between 'A, A'. 
obtained by 
constitute a 
The theo 
formed by t 
determinant! 
points, repla< 
consist in t 
“ normal ord 
solido may b 
such that tb 
are of unifoi 
pencil arran< 
a plane satisi 
privately prir 
(70) Hence we may contract at pleasure the given contour about any 
node as origin, and if the contour so contracted contains at least one node 
besides the origin, it will suffice to determine whether the given contour is or 
is not regular. 
theory of A.! 
against the i 
for the case c 
of invariance 
normal arran 
Thus for example in the case of a triangle limited by the axes and by the 
right line x + y = n, we may make n = 1 and the trial series will then become 
j | ^ which possesses the Farey property. Hence this will hold good for 
a triangular boundary of any size and wherever the origin is situated: this 
includes the case of the ordinary Farey series when the origin is taken at 
either extremity of the hypothenuse. So again for the area contained within 
the axes and the hyperbola xy = n, we may take xy = 1 and the trial series 
is the same as before. 
every other. 
LIST 
PROFESSOI 
Page 5, 
„ 6, 
» » 
(71) It is easy to form unperforated areas of any magnitude which shall 
not satisfy the Farey law: for example we may as in the annexed figure 
draw a curve passing through the origin, the point (0, 1), and the point (2, 3), 
0 2 
-, - does not satisfy the Farey law, and consequently no similar contour 
1 3 
obtained by treating any one of the three nodes which it contains as a centre 
of similitude will be a “ complete contour,” and the successive values of (p, q) 
„ 11, 
„ 13, 
t* Vol. 
[t Thei 
S. IV.