of Edinburgh, Session 1884 - 85 . 339 
which consists of the five triangles formed in the same way as 
above. 
So if a hexagon, heptagon, &c., be constructed in like manner, 
the line drawn necessary to complete any figure always intersecting 
the last drawn side of the preceding figure and the line 1, we shall 
have corresponding expressions for the areas of the figures formed 
in exactly the same way, and consisting of a series of terms denot- 
ing the areas of the triangles formed by either axis and the lines 
1, 2, 3, 4, &c., taken two at a time in order all round the figure, 
each line or number occurring twice ; so that if the area of a polygor 
of n - 1 sides be 
(12) + (23) + (34) + (45) + ... + (»■ *-2,»-l) + (7i + l,l), 
drawing an nth. line so as to cut the (n - l)th line and the line 1, 
the area of the polygon of n sides so formed will be equal to 
Area of polygon of n - 1 sides - A(l, n , n — 1) , 
that is, 
(12) + (23) + (34) + (45) + . . . + (ti - 1, n) + (tiI) , 
in which each number from 1 to n, and so each c and m in the 
equations to the sides, occurs twice, thus showing the truth of the 
law for a figure of any number of sides. 
As an algebraical result we may note : — We have seen that the 
sum of three expressions, involving any three suffixes, of which 
( c i c 2) 2 • 
m, - 
is a type, is equal to the sum of three expressions involving 
(tYI C TYt C 
the same suffixes, of which — — . is a type, each sum being 
mgrn.2 ( m 1 — m 2 ) J r 
A 2 
equal to the same expression of the form It follows also from 
the preceding results, that the sums of 4, 5, 6, . . . , n expressions 
(selected as indicated above), of the first type are equal to the sums 
of the corresponding number of expressions of the second type, each 
sum denoting twice the area of the polygon of the corresponding 
number of sides — the truth of which may also be shown algebrai- 
cally, and without reference to the areas of figures. For, completing 
the series of terms in any case by replacing those terms which 
cancelled, the expression for the area of a figure will consist of a 
