March 3, 1899.] 



SCIENCE. 



307 



scale. The greatest triangle has sides of 

 133, 167 and 190 miles (211,269, and 306 

 kilometers). The highest station is over 

 14,000 feet (4,267 meters). Operations of 

 such magnitude justify the introduction of 

 refinements not usually employed. The 

 latitudes are corrected for elevation, and 

 the horizontal directions are changed, to 

 reduce them to the sea level of the observed 

 station. A distinctive feature in the final 

 adjustment is the application of weights de- 

 pending on both the station errors and 

 those arising from the closing of figures. 

 These are treated separately, but the final 

 weights consist of two parts, one resulting 

 from local conditions and varying with each 

 direction, and the other deduced from the 

 formation of triangles and remaining con- 

 stant for the network under consideration 

 In the California work the probable error 

 of a direction at any station was 0".081 

 while that from the closing of triangles was 

 about twice as much. The latter neces- 

 sarily includes the former. The two are 

 separated by means of the formula 



which gives the resulting combination error 

 as ±0".169. e, is the probable error of a 

 direction from the closing of triangles, and e^ 

 is the average probable error of an observed 

 direction from station adjustment. Each 

 direction, therefore, enters the final adjust- 

 ment with a weight derived from measures 

 at its own station, added to the above value, 

 which represents the constant pai-t for the 

 entire figure. The cost of the transcon- 

 tinental arc from Cape May to San Fran- 

 cisco was two hundred dollars per linear 

 mile (S124 per kilometer), three and a half 

 dollars per square mile ($1.35 per sq. kilo. ), 

 and two thousand dollars per station. 



A fine example of rapid expansion from 

 the base to a fully developed net of triangu- 

 lation is found in the vicinity of Salt Lake, 

 and is a characteristic specimen of primary 



work as carried out in the Rocky Mountain 

 region of the United States. The average 

 height of the thirteen stations composing 

 the main scheme is 11,256 feet (3,431 

 meters) , while the average length of the lines 

 connecting them is 159,734 kilometers (99-|r 

 miles). The distance between Mt. Ellen 

 and Uncompagre is 294,104 kilometers (182f 

 miles). This remains to the present day 

 the longest line observed from both ends 

 and forming an integral part of a regular 

 system of triangulation executed by any 

 trigonometric survey in existence. Indeed, 

 the entire chain from the Sierra I^evada on 

 the west to the Mississippi plateau on the 

 east is without a parallel in similar work, 

 when we consider the magnitude of the 

 geometrical figures, the elevation of the 

 stations and the refinement of the individual 

 measures. 



Eeferring to a part of this work — the base 

 net at Salt Lake, Utah — the following de- 

 tails are of interest : 



The elevation of the base above sea level 

 is about 4,224 feet, while the mean height 

 of the stations composing the quadrilateral 

 is 11,088 feet. In only five steps we pass 

 from a base 11.2 kilometers in length to a 

 line 237,765 kilometers long (Pilot Peak, 

 Mt. Nebo). This involves an average mul- 

 tiplication of 4J times for each step of ex- 

 pansion, which is within the limit set for 

 development in well conditioned triangula- 

 tion. The resulting quadrilateral in which 

 the base line expansion culminates and on 

 which the transcontinental extension rests 

 contains nearly 10,500 square miles and is 

 the largest yet realized. The base net in- 

 cluding this figure (Ogden, Mt. ISTebo, Ibe- 

 pah and Pilot) was adjusted separately 

 and brought out the following criteria of 

 accuracy : 



FROM STATION ADJUSTMENT. 



Average probable error of a single observation 



of a direction =0".71 



Average probable error of an adjusted di- 



