December IS, 1S93.] 



sciENee. 



3^4 



In the original band we could pasa continuously along 

 the edge from B to A {^= D) to C (= B) back to our start- 

 ing point. Now, in cutting along the line EF we nowhere 

 cross this continuous edge, so that it will remain an edge 

 of the new surface, while the cut will form a second edge 

 of the new surface. We have thus removed one of the 

 distinctive characteristics of the double surface, by having 

 a surface bounded by two lines. Furthermore, it will be 

 impossible to pass in the new band from a point on one 

 side of the paj)er to the corresponding point on the oppo- 

 site side without crossing the edge. The reason for this 

 will be more obvious in the following: 



In forming the double service we revolved one end of 

 the strip of paper throvigh an angle of 180*^, about, say, 

 the point E, or, better, the line EF. It is clear now that 

 if, instead of revolving it through an angle of 180'^, we 

 had revolved it through an angle of 2x180*^, the point B 

 would have fallen on A and D on C, so that we should 

 have two continuous edges to our surface, and it will no 

 longer be a double surface. We can then say, in general, 

 that if we revolve the end of the strip of paper through 

 an odd multiple of 180*^, before fastening the two ends 

 together, we shall alwaj's obtain a double surface, whereas 

 if we revolve through an even multiple of 180^ we have 

 an ordinary surface. By this means we can distinguish 

 double surfaces from ordinary surfaces. 



We will now return to the siirface obtained by cutting 

 the double surface along the line EP and see why it must 

 be an ordinary surface. The double surface originally 

 had a twist of 180*^. Suppose, now, we have cut it as indi- 

 cated, but do not let the ends drop apart; then each part 

 on either side of the cut will have a twist of 180*^, or, to- 

 gether, 2x180'". If we let the surface fall apart we double 

 the twists again, and our new surface has a twist of 4x180'^, 

 and it is therefore an ordinary surface. That this is so 

 may be easily verified by cutting the band across and re- 

 volving one end until the strip has no twists, when it will 

 be found that it has to be revolved through 4x180°. 



If we cut a double surface obtained by rotating thi'ough 

 3x180'^ along the line EF, we shall find that we have intro- 

 duced a knot in our new surface, which in other respects 

 will, however, be an ordinary surface. These knots will 

 be multiplied, as we proceed, to surfaces containing a 

 higher number of twists. 



It is easy to see that if we cut an ordinary surface, ob- 

 tained by revolving the end of our paper through 2x180", 

 along the line EF, we shall obtain two ordinary surfaces, 

 which are, however, interlinked. The same holds for 

 surfaces with a higher number of twists, where, however, 

 the interlinking becomes more complicated. 



Another interesting set of results may be obtained by 

 cutting the surfaces along a line parallel to the edge at a 

 distance from the edge less than one-half the width of 

 the strip of paper. The results will be different in the 

 case of double and ordinary surfaces. 



BOTANICAL NOTES FROM WESTERN PENN- 

 SYLVANIA. 



M\ HUBERT LYMAN CLARK, PITTSBURGH, PA. 



On looking- over my field notes for the spring- and 

 summer of 1893, I find there are a few facts the preser- 

 vation of which may be worth while, in the hope that 

 before long some competent botanist will prepare an 

 annotated list of the plants of western Pennsylvania. 

 There is not at present, so far as I know, any such list, 

 and its appearance would be welcomed by all our local 

 botanists. Whenever the work is undertaken it will be 

 desirable to have as much material in available form as 



possible, and so I have ptestlmed to publish my im- 

 portant notes in Science, hoping they will also prove of 

 interest to botanists elsewhere. 



On analyzing specimens of Delphinium from the covm- 

 try around Pittsburgh last spring, I was struck with the 

 rich coloring- of the flowers. There was not the least 

 doubt about the plant being D. tricome, but, to my sur- 

 prise, Gray's "Manual" says the flowers of X*. tricome are 

 "bright-blue, sometimes white," while every specimen 

 which I examined had "royal purple" flowers. Think- 

 ing that the trouble might be in my sense of color, I 

 loMied through the "Manual" for other "bright-blue" 

 flowers. I found Airier undulatus, Chicorium intybus and 

 Campanula rolundifolia so given, and I should certainly 

 call them so, but the Delphinium of this vicinity has 

 flowers of the same color as Lialris scariosa, which Gray 

 calls "rose-purple," or perhaps nearer to Anter novce- 

 anglice; which is given as "violet-purple." Never having 

 seen Delphinium growing elsewhere, I am curious to 

 know if in other parts of its range it really does bear 

 flowers similar in color to Chic.rium or Campanula, or 

 whether it is not a slip of the pen in the "Manual" to 

 describe them as "bright-blue." In the same work 

 (which is perfectly invaluable to an amateur botanist in 

 the east) Silene nicea is recorded as "rare," and it is with 

 great pleasure, therefore, that I can report it as abund- 

 ant in several places around Pittsburgh. Indeed, I am 

 inclined to think it is the most common representative 

 of its genus in this neighborhood. 



None of the botanists whom I have consulted record 

 Trifolium doloniferuvi east of Ohio, and it is therefore 

 very pleasant to be able to record it from Pittsburgh. 

 On the 8th of last June I found it growing in an open 

 space in some woods about six miles east of the city. 

 While it is of course possible that it has been introduced, 

 it was growing so far from any house or highway as to 

 certainly appear indigenous. There is no specimen of 

 this clover in the herbarium of the Western Pennsyl- 

 vania Botanical Society, and I am inclined to think this 

 is the first record for the State. 



Gnaphalium purpureum is reported in the "Manual" to 

 occur in "sandy or gravelly soil, coast of Maine to Vir- 

 ginia and southward." It is not very clear from this 

 how far inland we may expect to find it, but certainly 

 the implication is that it is a seashore plant It may be 

 worth while, therefore, to record that it is not very rare 

 around Pittsburgh, three hundred and fifty miles from 

 the coast! I found it growing in Arlington, twelve 

 miles south of the city, in June last, and there are a 

 number of specimens in the herbarium of the botanical 

 society to which reference has been made. These speci- 

 mens are from widely-scattered points in the county, 

 and would seem to indicate regular and not uncommon 

 occurrence. One of the most abundant weeds in many 

 parts of Pittsburgh is a species of Galinaoga, differing 

 from G. parvijlora in the scales of the pappus and being 

 very hairy instead of smoothish. Dr. Robinson, of the 

 Gray Herbarium, Cambridge, to whom I am indebted 

 for man)^ favors, kindly identified the specimens sent to 

 him as "a possible variety of G. hispida," or at least so 

 it may be considered provisionally. Similar specimens 

 are reported from Milwaukee and Providence. Whether 

 G. parviflora occurs in Pittsburgh I cannot say, but I have 

 not yet found any specimens agreeing with the descrip- 

 tion in the "Manual. " Another plant which Dr. Robinson 

 identified for me is also an introduced species reported 

 in the "Manual" as "ratherrare;in cultivated grounds." I 

 refer to Veronica arvenais, two specimens of which I sent 

 to Cambridge, supposing them to represent different 

 species, they were so unlike. One of them was collected 

 in open pasture land, and I found similar specimens in 



