162 Appendages of Trilohites. 



dips, and the method will be found to have some advantages over 

 that given above (iii.)- 



Using Fig. 10, from Z draw ZA and ZB in the directions of the 

 observed dips, but of lengths representing their complements; draw 

 AB and ZD perpendicular to it; then ZD is the direction of full 

 dip and corresponds in length to the complement of that dip. The 

 edge of the protractor must be applied to ZD, and the reading thus 

 obtained subtracted from 90 degrees to obtain the true dip itself. 

 The proof of this is evident from the first of the equations (a). The 

 method is equivalent to that of Mr. Hill cited above. 



All the above constructions, with the exception of the second (me 

 in problem (xi.), are theoretically exact: the accuracy actually 

 attained will of course depend upon the precision with which the 

 drawing is performed. In some cases a small error in the drawing 

 may give rise to a large error in the results, but this is due not to 

 any fault in the method, but to the inadequacy of the data. For 

 example, in problem (vi.), Fig. 9, if the slope of the ground in the 

 two localities be nearly the same, and the directions of outcrop 

 nearly the same, the lines EA and FA will meet at a small angle, 

 and any error in drawing them will produce a magnified error in the 

 position of A, and therefore in the direction and magnitude of ZA ; 

 but this is because the observations are insufficient to determine the 

 dip of the beds with any degree of accuracy. 



The foregoing examples are enough to illustrate the wide appli- 

 cations of graphical methods ; solutions of other problems on similar 

 lines will suggest themselves. 



III. NOTKS ON THE APPENDAGES OF TkILOBITES. 



Note to accompany Three Woodcuts of Asaphits megistos, a Trilobite 

 discovered by Mr. James Pugh, near Oxford, Ohio, in the upper 

 portion of the Hudson Eiver Group. 



WE are indebted for the Woodcuts accompanying this note to 

 the courtesy of Dr. John Mickleborough, whose jjaper on the 

 Locomotory Appendages of Trilohites we published in our February 

 Number, p. 80. 



They serve admirably to confirm the observations of Mr. E. 

 Billings (published in the Geological Magaztnk for 1871, Yol. 

 VIII. PI. VIII. pp. 289-294) on the appendages of AsapJws plaiy- 

 cephalus. In Fig. 1, a, a, mark the position of the anterior pair of 

 appendages ; h, h, the 10th pair ; c, the articulation between the 

 carpus and propodos ; d, the articulation between the propodos and 

 dactylus ; e, the lines to the letter e mark the position of the lamelli- 

 form ? branchiae beneath the pygidium. 



Fig. 2 represents the upper surface of specimen reduced to nearly 

 one-third natural size. 



Fig. 3 shows the mould into which Fig. 1 fits, a, a, mark the 

 moulds of the bases of the anterior pair of appendages, h, b, of the 

 10th pair of appendages ; the lines leading to c inclose the probable 

 space to which the lamelliform ? branchia3 were attached ; d marks 

 the position of the left maxillipede ; e the left angle of the hypostome. 



