862 



SCIENCE. 



[N. S. Vol. XXI. No. 544. 



Professor O. H. Basquin : ' The Bending 

 Moment of a Uniformly Loaded Beam; a New 

 Experimental Demonstration.' 



Floyd Field, 

 Secretary. 



DISCUSSION AND CORRESPONDENCE. 



CONNECTION BY PRECISE LEVELING BETWEEN THE 

 ATLANTIC AND PACIFIC OCEANS. 



To THE Editor of Science: In your issue 

 of April 28, 1905, page 673, is an article by 

 llr. Hayf ord on ' Connection by Precise Level- 

 ing between the Atlantic and Pacific Oceans.' 

 About twenty years ago I wrote to Science in 

 connection with precise leveling over the Alle- 

 ghanies and the Kocky Mountains, and stated 

 that it might be well to have a systematic 

 determination of bench marks at stated in- 

 tervals owing to the unrest in the earth's crust. 

 At that time I stated that my work on the 

 corps of the Pennsylvania Railroad had shown 

 me that, however carefully the bench marks 

 might be established at any one time, at the 

 expiration of a comparatively few years there 

 would be a discrepancy between them and the 

 datum plane. The Pennsylvania Railroad has 

 reviewed its bench marks a nmnber of times 

 owing to these discrepancies due to earth mo- 

 tion. The want of agreement, therefore, be-^ 

 tween the levels of the Atlantic and Pacific 

 Oceans, unless the bench marks were estab- 

 lished by surveys which began and ended at 

 exactly the same period throughout the entire 

 distance, might be due to earth movements 

 between the times of the beginning and the 

 end of the survey. 



I would again suggest, as I did at my first 

 letter to this paper, that the United States 

 Geological Survey secure not only the lists of 

 bench marks of all railroads, but the varia- 

 tions that have occurred in these bench marks 

 as shown by repeated surveys. If these are 

 carefully tabulated throughout a century, we 

 may obtain important information in regard 

 to the upward and downward crustal move- 

 ments across the continent. 



Edwaiu) it. Williams, jr. 



SPECIAL ARTICLES. 



the horizontal plane of the skull AND THE 

 GENERAL PROBLEM OF THE COMPARISON OF 

 VARIABLE FORMS. 



In comparative studies of the skull it is 

 customary to select one transversal plane de- 

 fined by the axis of symmetry with which it 

 is at right angles and by two points, as the 

 standard plane to which the skull is referred. 

 Some authors have made the selection of the 

 two determining points based on morpholog- 

 ical considerations, while others have en- 

 deavored to determine the physiological hori- 

 zontal position, determining the latter by two 

 points which are more or less accurately par- 

 allel to the direction of horizontal sight. 



When this problem is considered from a 

 purely morphological point of view, it will be 

 recognized that there is no justification in 

 selecting arbitrarily two points and disregard- 

 ing all others, but that the best method of 

 comparison must be based on the assumption 

 that every point of the skull has equal weight 

 and that the nearest approach of all points 

 must be attempted. In this form the problem 

 is applicable to the comparison of all variable 

 forms. 



The most favorable superposition of any 

 two forms will be obtained when the sum of 

 the squares of the distances between all pairs 

 of homologous points becomes a minimum. 

 We will refer the body to a system of rect- 

 angular coordinates and call x', y' and z' the 

 ordinates of a point of the first body, x", y" 

 and z" the ordinates of the homologous point 

 of the second body. By moving the second 

 body by the amounts u, v and w in the direc- 

 tion of the three ordinates, we can modify the 

 relative positions of homologous points with- 

 out torsion of the body. Then the sum of the 

 squares of the distances of homologous points 



2(a' — i"— u)2-|- j"— J^)'^-f 2(2'— 3"— 'f)^ 



is to be a minimum. Therefore, 

 2(a:' — a/' — u) =0. 



And 



