ON SOME TRACKS OF INVERTEBRATE ANIMALS. 19 



By addition of these four equations we obtain 



(A^)^ + (A^)^ + ( A^)^ + (A^,)^ = K{m^ + 2m^r + 1^) 



From this it seems likely that if we repeated the operation 

 n times the aggregate of the 2'' areas obtained would be 

 equal to the primitive area, and it may be readily shown 

 that if the proposition be true after n operations,, it will be 

 true after n+i operations. But it has been shown to be 

 true when n is equal to 2, therefore when n is equal to 3, 

 therefore when n is equal to 4 &c., and so the proposition 

 is generally true. 



III. On some Undescrihed Tracks of Invertebrate Animals 

 from the Yoredale Rocks, and on some Inorganic 

 Phenomena, produced on Tidal Shores, simulating 

 Plant-remains. By Professor W. C. Williamson, 

 LL.D., F.R.S., President. 



Read February loth, 1885. 



[Plates I., II., III., & III'.] 



About two years ago I received from the Rev. Isidore 

 Kavannah, of Montreal, then a student of Stonyhurst 

 College in Lancashire, some interesting objects which he 

 had discovered upon some loose blocks of stone strewing 

 the shore of the river Ribble, close to the College. The 

 raised bank of the river, at that point, consists of hard 

 beds of Yoredale rock separated by thin layers of softer 

 material. A careful examination of the locality left no 

 doubt on my mind that the specimens had fallen from the 

 under surface of one of these hard beds. Though we failed 



c2 



