498 REPORT— 1891. 



then on moving the rod from A to C the roll will be negative. We there- 

 fore say that the area generated is likewise negative. 



This shows that we must also give the rod QT a definite sense, and we 

 shall take this as positive from Q to T, so that Q is the beginning, _T the 

 end of the rod. A reversion of this sense is equivalent to a reversion of 

 the graduation of the wheel. To fix the idea, I shall suppose that on looking 

 along the rod in its jwsitive sense— i.e., from _Q to T — the numbers on the 

 graduation of the wheel i7icrease in the clockwise sense. 



Under this supposition the line QT, when placed anyhow on the paper, 

 •will generate a positive or negative area according as it is moved to the 

 left or to the right respectively. This can be put differently. 



Rule for Sign of Area. 



Let the rod, or generating line, QT jmss over a point P ; then the area 

 near P will be j)ositive if to a j)erson standing at P and looking along the 



jwsitive sense of the rod the latter moves 

 I Fig. 2. from his right hand to his left. The 



area irill be negative if this motion is 

 from left to right. This rule holds quite 

 independently of the length of the gene- 

 rating line. In fact, as it is stated, a 

 small piece of the line in the neighbour- 

 Jiood of the point which crosses P has 

 alone to be considered. The rule will 

 therefore also hold if the line should 

 vary its length. 



More than this. Let the line turn 

 about Q as a fixed point from QT to QT' (fig. 2). It will now sweep over 

 the sector of a circle whose area is fj QT^ii if 6 denotes the angle of turn- 

 ing. A wheel W at the distance c from Q will record a roll tc^^c'K If 

 QT:=l, we get, therefore, 



Area generated =^ l'^0=h l^ — 



" (; 



The roll of the wheel will thus again record the area. This will be 

 positive for counter-clockwise turning, negative if the turning is clockwise. 



If the wheel should be mounted on the continuation of QT beyond Q, 

 then c will be negative ; so will be the ' roll,' and the result will be posi- 

 tive again. It will thus be seen that our rule holds also for this case. 



First Mode of Generating an Area. 



Consfder now fig. 3. Let the lineQT start at AA' and move to BB', 

 the point Q remaining always on OX and QT perpendicular to the latter, 

 whilst T moves along the curve A'B'. The line QT will now generate 

 the area ABB'A', but in doing so it will continuously change its length. 



With regard to the sense of the area the rule of sign holds. Hence 

 if, in fig. 4, QT moves from AA' to BB', whilst T remains on the lower 

 branch of the curve, it will sweep over the area AA'C'B'B, and this will 

 be negative. But if now the point T moves back to A' on the upper 

 branch, then the area BB'TA'A will be swept over in the positive sense. 

 The whole area generated will be the difference of these two areas — i.e., it 

 will be the area of the given closed curve. 



