the only necessary condition being that the total length of the lines, 

 straight or curved, shall vary with t as assumed in (1). The 

 essentially two-dimensional integral would then be represented by 

 the total surface generated by the motion of f(x) along the axis t y 

 overlapping, should it exist, being taken into account. 



Similarly if the generating function be essentially two-dimen- 

 sional /(x, y) say, it may be a plane surface, or a series of such of 

 any form whatsoever, provided only that its, or their, total varies 

 with t as expressed by the function A t ; and the function V t will 

 accordingly be represented by the volume — in right-cubic or paral- 

 lelepipedic units, according as the system of axes is rectangular 

 or oblique — of the path or paths of the former. Overlap, if exis- 

 tent, must as before be taken into account, which fact being quite 

 general need not be further referred to. 



Again, if motion in the axis t imply some variation in physical 

 condition 1 of the surface or surfaces A t , dependent merely upon 

 its or their total area, i.e. independent of the order of the surface 

 or surfaces, then it or they, though actually three-dimensional, 

 may be regarded as essentially two-dimensional. 



Similarly also, the function V t will represent the integrated 

 effect due to the motion along the axis t, or to rotation through 

 the angle t, or to the lapse of time t, of any three-dimensional 

 figure subject to such change of form or condition as may be 

 expressed by (1). Analogous illustrations will serve to elucidate 

 the nature of the integral, applied to space of higher dimensions. 



It is evident that not only all the regular forms ordinarily 

 considered in plane and solid geometry are included in the range 

 of the function, but also very many others, and that its applica- 

 tions are not restricted to ordinary or tri-dimensional space. 



3. Oblique rectilinear axes. — If the axis x, of the generatrix, 

 and the axis t, be not orthogonal, the units of the generated sur- 

 face will be oblique, hence if x make an angle w with the plane 

 perpendicular to t, its projection-length thereon will be cos to, 



1 E.g. variation of intensity, see Journ. Eoy. Soc, xxxv., pp. 284 - 286. 

 U— Dec. 4, 1901. 



