98 SPOTTISWOODE ON TYPICAL MOUNTAIN RANGES. [April 23, 1860. 



correction afforded by the calculations in tlie present instance, as 

 to suggest and exemplify a new problem in physical geography. 



He accordingly examines one of the groups of Dr. Abich, consist- 

 ing of 24 mountains. He first finds the mass of each from the data 

 of their length, of their elevation, and of the slope of their sides, 

 which latter, in absence of better information, he assumes to be the 

 same in every case. Now this mass corresponds, in mathematical 

 language, to the ' weight ' of an observation, and is represented by w. 



Next, he takes Oy , 6^ ^„ as the angles between the several 



mountain ranges and the parallels of latitude, drawn through the 

 middle points of the ranges ; in which case, by well-known formulae, 

 the probable typical direction of the chain will be 



2 WJ0 



e = 



Moreover, the probable errors (or deviations) of the various 

 ranges from the typical range will be ^i = - ^i . . . . e„ = © - ^„. 



The mean error (deviation) of an observation (range), whose 

 weight (mass) is unity, will be 



/Yw^ 



The mean errors of ^i, 6^ 6„ respectively, or the " errors (de- 

 viations) to be feared " will be 



a/ Wi V 102 ^ '<^n 



And the mean error E (deviation) of O, or " error (deviation) to 

 be feared " will be 77 



So that will lie between the limits G db E- 



An ample numerical investigation is made of all these quantities ; 

 they are tabulated, and they are compared with a theoretical scale 

 of precision in the way well known to all who are familiar with the 

 subject of probabilities. The result is, that not only has the mean 

 direction of these mountain masses been accurately ascertained, but 

 it is also made clear that the direction in question is more than a 

 mean direction — that it is a typical one. In other words, we have 

 a direct numerical value (and it is in this case a large one) for the 

 probability that the directions of the different mountains forming 

 this group are due to a single cause, and not to many independent 

 causes; and, consequently, the geologist and the physical philo- 

 sopher will have good grounds to go upon in seeking some common 

 agency which has caused their upheaval. 



