96 Mr. J. Nixon's Heights of Whernside, Great Wkernside, 



expresses himself as follows : " La theorie des equations de- 

 rivees, porte naturellement k conclure que toute valeur qui 

 peut satisfaire a une equation derivee donnee doit etre ren- 

 fermee dans son equation primitive, pourvu que celle-ci ait 

 toute la gencralite dont elle est susceptible, par les con- 

 stantes arbitraires qui doivent y entrer. II y a iioanmoins 

 des equations dcrivees auxquelles satisfont des valeurs que 

 j'appelie singtdieres, parce (ju'elles ne sont pas coi'iprises dans 

 leurs equations primitives*." Again: " On doit conclure de 

 la que, pour que x soit une valeur singuliere non comprise 

 dans la valeur gcnerale, il faut," &c.t 



I here terminate these observations; nor shall 1 again 

 trespass on the pages of the Philosophical Magazine, by any 

 further remarks upon a subject which has now been so fully 

 set before its readers. 



July 6, 1836. 



XXI. Heights oflVhernside, Great Whernside, Rumhies Moor, 

 Pendie Hill, and Boulsworth. By John Nixon, Esq-X 



I^HE following trigonometrical differences of level, mea- 

 sured at numerous stations for the determination of the 

 above altitudes, having been calculated, under a range of di- 

 stances of from four to thirty miles, with the constant refrac- 

 tion of the formula given in my last (vol. viii. p. 4-80), are sub- 

 mitted as a severe test of its claims to general accuracy. The 

 details of some of the measurements have been already given, 

 and those of the remainder will appear in my surveys of 

 Wharfdale, Ribblesdale, &c. 



For every station each day's observation of the difference 

 of level between the standard hill and the other (calculated in 

 the manner described in pages 437-8 of vol. vi.) is arranged 

 in a separate line. 



Trigonometrical Differences of Level. 



Whernside above Inglcborough. 

 Feet. 

 At Great Whernside .. .. 45 



Scttronside\*i 405 («) 



1 41-2 



Raisegill Hag .. .. 41-8 

 Cosh 40-6 



Feet. 

 Atlngleborough .. .. 41 '2 



38-8 



400 



Penvgent 42-3 



1^ 40-9 



39-8 



* Calcul des Fonctions, p. 1/8. t ^^i'l; P- 234. 



X Communicated by the Author. 



§ The niiiik (n) dviiolcs tliat as there was no accoiniianyiiig ol)scrvation 

 of ttie standard hill, its avcnigc diiicrcnce of level was iibcd in the calcula- 



tion. 



