TRANSACTIONS OF SECTION A. 537 



Hence the area included in A and not in B will be positively generated. In like 

 manner the area included in B and not in A will be negatively generated, while 

 the area common to A and B, or exterior to A and B, will disappear from the 

 integral. We therefore have 



B-A= lp,„d(f), 



in which the area B or A is to he regarded as negative if its perimeter is described 

 in the negative direction. When BA is a line of fixed length, let I = 2b, then 



B-A = 26L,„# .... (1). 



2. This theorem may be employed in the explanation of Amsler's Planimeter ; 

 for let OA = a be the bar which rotates about the fixed point O, and let AB = 26 

 be the bar carrying the recording wheel, situated, say, at the distance c from its 

 middle point (see figure). Then, s denoting the distance recorded by the wheel when 

 B describes the perimeter of a closed curve. 



( = (p + c) <f<^ = pmJ<t^ + ^Jcttc, 



in which k denotes the number of revolutions made by the bar AB. Hence 

 from (1) 



B-A = 2bs-4kTrbc. 



In the use of the instrument two cases arise. First, 

 may be exterior to the area B ; in this case k = 0, and 

 since the point A reciprocates over an arc of a circle, 

 A = 0, and we have 



B = 2bs. 



Secondly, may be within the area B, in which case 

 A = Tra- and k=l, hence 



B = 2bs + Tr{a^-4bc). 

 The constant which is added to 26s in this case is the area of the circle whose 

 radius is OB, when the instrument is in such a position that the line joining O 

 with the wheel is perpendicular to AB ; for OB- = a' — {b + c)'^+{b — c)^; and this 

 should be the case, for if B describe this cu'cle s is evidently = 0. 



3. The theorem (1) may be employed also to express the area described by any 

 point of the line AB in terms of the areas of A and B. Thus, if s denotes the 

 distance recorded by a wheel situated at the middle point of AB, 

 B-A = 26s (2). 



Let C be a point at a distance c from the middle point of AB. The length of 

 AC is b + c, and the distance of its middle point from the middle point of AB 

 is — ^ (6 — c) ; therefore the distance recorded by a wheel situated at the former 

 point would be 



s' = 8-k7r{b — c), 



(k denoting as before the number of revolutions of AB), and we have 



C-A = {b + c)s' = {b + c)s-kTrib^-c^) . . (3). 



Eliminating s by means of (2), 



C = ^-^A + ^-^^:B-kn{b"--c^) . . . (4), 



or, if 26 = i and ; — = — , we have for the area described by the point which cuts 

 ' b-c n 



AB in the ratio m : n 



^ «A + wB kmn ,, 



C = -J n l\ 



m + n (9)1 + n) 



