28 The Rev. D. Williams's Cliff Section of Lundy Island. 



in which we are at liberty to suppose^/' to be the roots of 

 a + c/'^=0, which reduces it to the form 



_„ d'^u T), ^M 



~ dxdz dj/ 



which may be further reduced, as before, to the form (4.). 



Thus it has been proved that (1.), (2.), (3.)j (4-) are the fun- 

 damental forms of partial differential equations of the second 

 order with constant coefficients, for two and three variables. 

 If there be more variables, it is evident the method here em- 

 ployed will still apply, and enable us to reduce them to fun- 

 damental forms. I have not had occasion however to pursue 

 the method beyond what is here done; nor have I applied it 

 to equations of the third and higher orders, though I am in- 

 clined to think that with some modification it would enable us 

 to reduce them to fundamental forms. 

 Sheffield, May 18, 1849. 



III. Cliff Section of Lundy Island, from the Sugar-Loaf to the 

 DeviPs Limekiln. By the Rev. D. Williams*. 



THE specimens to which the following descriptions refer, 

 were collected and numbered on the spot, and occupy, 

 as near as may be, the relative positions assigned them on the 

 section. 



No. 1. Granite, often of the porphyritic variety, a ternary 

 compound of mica, quartz, and felspar only. 



2. Gray granite, differing only from the former by the ac- 

 cession of rare and remote specks and crystals of foliated 

 hornblende, which apparently replace the mica to the amount 

 only of such accession. 



3. Gray granite, more syenitic, of a greenish tint, not so 

 distinctly crystalline granular as the last, at times calcareous, 

 and the mica in a greater degree absent. 



4. Gray granite, with a less shade of green, no hornblende, 

 slightly calcareous, mica none or scarcely perceptible, the 

 quartz crystalline granular, and the felspar in well-defined 

 and confusedly crystalline arrangement. 



5. Gray, compact, or finefy granular granite rock, with no 

 trace of hornblende, the mica absent or extremely minute and 

 scarcely distinguishable, but otherwise compounded of felspar, 

 quartz, carbonate of lime, and calcareous spar ; the two latter 

 in such augmented proportions, as to cause the rock to effer- 

 vesce briskly or feebly at nearly all points on being tested by 

 acids. 



* Communicated by the Author; being the substance of a memoir 

 communicated to the British Association, &c. held at Swansea in 1848. 



