4 SCIENCE PROGRESS 



the set of points inside the curve, with or without the points 

 of the curve itself. 



But the formula 



— / xdy —ydx 



which thus gives the area when it exists, leads, when the in- 

 tegral is taken round a looped curve, only to the sum of the 

 areas of the loops. Area is, in fact, a directed quantity — positive 

 when the point describing it has the inside of the loop on its left. 

 The most important matter, in which more attention to 

 directed area is needed, arises in the transformation of the 

 variables in a multiple integral. In such transformations, as 

 ordinarily presented, the variables are regarded as steadily in- 

 creasing or decreasing — a supposition which fails in relation 

 to functions of bounded variation, which will not usually be 

 monotone or of uniform sign. Prof. Young takes up the 

 problem of establishing the formula 



ffdxdy =ff%^. dudv 



d{u, v) 



under conditions which have no reference to the sign of the 

 Jacobian, or even of the partial derivates, area being suitably 

 defined. We may quote the author's suggested definition. 

 Inscribe in a curve, supposed to be closed, a set of polygons as 

 usual, of which the perimeters approximate to the length when 

 the curve is rectifiable. Imagine vectors to be represented in 

 magnitude, sense, and line of action by the sides of the polygon, 

 supposed described in the sense in which a parameter, u, in- 

 creases. Take the moments of these vectors about any point 

 Oin the plane. Then their sum is independent of 0, and equal 

 to twice the area of the polygon as usual when the polygonal 

 line does not cut itself. Moreover, the sum has a unique limit 

 2A when the number of sides of the polygon is increased, so 

 that the perimeter is the length- of the curve if this be recti- 

 fiable. We define A as the area of the curve. 



With this definition, complete precision appears to be 

 attained in the statement of all the fundamental results of the 

 calculus relating to areas. We shall not, however, quote the 

 necessary modifications in their statements. The author has 

 done considerable service in this exposition of some of the 

 difficulties which even elementary students must have felt in 

 regard to this rather neglected subject. 



W. P. Milne and D. G. Taylor (part 5, p. 375) discuss the 

 significance of apolar triangles in eUiptic function theory. Much 

 work has been done on this subject recently, but apolarity has 

 not been applied to a great extent to curves which are not 



