140 Mr. Nixon on the Measurement (by Trigonometry) of the 



denote the angles which are formed between c y, c' y', c" y", &c. 

 and c b, c' b', c" b", &c. respectively, 2 m .cb .cy . cos & is either 

 = or negative. Now we have yb 1 = cb 2 + c y ' — 2 . e b . 

 c y cos £ ; and it is clear that 



2 m , . yb 2 — 2 m . cb 2 = 2 m .cy — c 22 .m.cb.cy. cos 3 



and is consequently always positive, and hence that 2m.yb 2 

 is always greater than 2m.cb\ or that 2 m . c b 2 will be a 

 minimum, g. e. d. 



It is very remarkable that the free motions of a system, when 

 not consistent with the necessary conditions of the same, are 

 modified by nature in the same manner as the calculating 

 mathematician, following the method of minimum squares, ba- 

 lances results which refer to magnitudes connected with each 

 other by a necessary dependence. This analogy might be 

 further followed up, but this does not lie within the scope of 

 my present design. 



XVII. On the Measurement (by Trigonometry) of the Heights 

 of the principal Hills of Swaledale, Yorkshire. By John 

 Nixon, Esq. 



[Continued from page 12.] 



TN the winter preceding the survey of Swaledale, some ex- 

 A periments, undertaken to determine on a novel plan the 

 cylindrical error of the horizon-sector, led to the rejection of 

 the original object-glass of its telescope for one of shorter 

 focus, but much superior in the centering. The first trial of 

 its merits took place at Great Whernside on a perfectly calm 

 and remarkably clear afternoon in May ; the test being the 

 consistency of repeated observations of the respective depres- 

 sions of the nearly level summit of Ingleborough and the 

 rounded one of Shunnor Fell; two different, yet equally dif- 

 ficult subjects for the correct pointing of the horizontal wire 

 of a telescope of a moderate power. On examining the an- 

 gles (corrected for the deviations of the bubbles from their re- 

 versing points,) it was remarked that the depression of Shun- 

 nor Fell had diminished as the decidedly frosty evening ad- 

 vanced, the decrement amounting to twenty seconds, whilst 

 the nearly contemporaneous depressions of Ingleborough might 

 be considered, one measurement excepted, to have been con- 

 stant. Now as the Swaledale observations, compared with 

 those of the preceding surveys, present an unusually limited 

 range in the corresponding errors of collimation, the superi- 

 ority of the present object-glass over the former one may be 

 concluded to be satisfactorily established, and the horary di- 

 minution 



