GEOMETRICAL INSTRUMENTS AND MODELS. 37 



tion depends entirely on the properties of situation of the two 

 bounding surfaces; and it is one which has been found to be of 

 some importance in the theories of the motion of fluids, and oi 

 electricity. 



As a third example, take an oblong strip of paper, and fasten its 

 two ends together so as to form a portion of a cylindrical surface. 

 Take another similar piece of paper, and again fasten its two ends 

 together, but give the paper a half twist so as to bring the upper 

 surfaces of the two ends in contact with one another. Between 

 the surface thus formed, and the cylindrical surface at first 

 obtained, an important distinction will bte found to subsist ; viz., 

 the cylindrical surface has an outside and inside surface, and there 

 is no way of passing from one to the other except by penetrating 

 the paper or crossing its edge : whereas the two sides of the 

 second surface form one perfectly continuous sheet ; so that by 

 travelling once along the whole length of the oblong strip, we 

 should pass from a point on the surface to the point exactly cor- 

 responding to it on the other side of the surface ; and we should 

 not return again to the point from which we set out, until 

 we had completed the tour a second time. The distinction 

 which we thus learn to draw between surfaces which have two 

 sides, and surfaces which have but one, is fundamental, and 

 depends solely on the properties of situation of the figure, as we 

 have now defined them. 



No complete corps de doctrine has yet been formed of the pro- 

 perties of situation of figures. This is partly owing to the great 

 difficulty of the inquiry, partly to the fact that it is only in very 

 recent times that the' attention of mathematicians has been 

 called to the subject, by the unexpected light which researches 

 into it have been found to throw on some of the most obscure 

 questions of the integral calculus. We cannot, therefore, expect to 

 find this part of the science of geometry extensively illustrated by 

 models, or by drawings expressly prepared for the purpose. But 

 any great collection of geometrical objects cannot fail to supply 



