l899-'00 TRANSACTIONS 153 



we have another illustration. It is stated that : ' ' from a geologi- 

 cal standpoint, but hardly from a practical one, however, there is 

 another reason why monuments should not be placed on the 

 summit of the Bitter Root range, as marking the boundary line 

 between Idaho and Montana. There is abundant evidence that the 

 summit is what is known as a retreating or migrating divide ; in 

 other words, the waters tributary to the Bitter Root River in 

 Montana are continually capturing by erosion those of the Clear- 

 water river in Idaho, so that the divide is slowly being shifted to 

 the westward, thus adding to the territory of Montana and 

 diminishing that of Idaho. The existing divide is uniformly from 

 six to eight miles from the irregular line representing the 

 original divide, if the latter may be accepted as having passed 

 through the highest points of the range, which seems probable." 



When a boundary is defined by a parallel of latitude, the 

 question invariably arises, in the demarcation of it, whether the 

 astronomic or mean parallel is to be adopted. The astronomic 

 parallel is that line on the surface of the earth on which direct 

 observations for latitude give the same elevation of the pole ; 

 geometrically, for the spheroid or ellipsoid of revolution, it is the 

 intersection of the cone, having its apex in the minor axis of the 

 earth and making an angle therewith equal to the complement of 

 the latitude, with the surface of the earth. Principally owing to 

 the local deflection of the plumb line, points astronomically deter- 

 mined in latitude will not " close," that is, the line projected or 

 run from one station as a parallel will not meet the next point or 

 astronomic station. 



That line with reference to which the sum of the dis- 

 crepancies north is equal to the sum of those south is the mean 

 parallel. 



However, as the latter can only be determined after the 

 location and connection of the astronomic points, entailing re- 

 vision of the whole work, and besides the difficulty of re-establish- 

 ing points on the mean parallel in case of loss or disappearance of 

 monuments and marks, it has generally been decided to adhere to 

 the simpler and more readil)^ established astronomic parallel. All 

 such parallels traced upon the earth are irregular curves. 



The International Boundary Ivine, between the I^ake of the 

 Woods and the Rocky (Stony) Mountains is defined in the second 



