157 



Mathematics. — " Com hina lion of observations nnth and without 

 conditions and detennination of the weights of the unknown 

 quantities, derived from mechanical principles. By Prof.' M. J. 

 VAN UvEN. (Communicated by Prof. Jan de Vries), 



(Communicated in the meeting of May 30, 1914), 



The theory of the combination of observations by the method oi 

 least squares has already been the object of numerous geometrical 

 and mechanical illustrations. In the geometrical representations the 

 leading part is usually played by vectors (L. von Schrutka ^), C. 

 Rodriguez ') ) ; the mechanical ones are taken partly froiii tlie theory 

 of the "pedal bary centre" (Y. Villarceau '), M. D'OcAOifJE ') ), partly 

 from the theory of elasticity (S. Finsterwalder ^) R. d'Emilio"), 

 S. Wellisch, Pantoflicek 0, F. J. W. Whipple '), M. Westergaard =•), 

 G. Albenga ^^)). 



In the following paper we will try to develop a mechanical 

 analogy of the solution of the equations furnished by observation, 

 supposing that no conditions are added, as well as for the case 

 that besides the approximate equations of condition (called by us: 



^) L. VON ScHRUTKA. Eine vectoranalytisclie Interpretation der Formeln der Aus- 

 gleichungsrechnung nach der Methode der Ideinsten Quadrate. Archiv der Math. u. 

 Physik, 3. Reihe Bd. 21 (1913), p. 293. 



2) G. Rodriguez. La compensacion de los Errores desde el punto de vista geo- 

 metrlco. Mexico, Soc. Gientif. "Antonio Alzale", vol. 33 (1913—1914), p. 57. 



3) Y. Villarceau. Transformations de I'astronomie nautique. Gomptes Rendus, 

 1876 I, 531. 



■*) M. d'OcaGiNe. Sur la determination géométrique du point Ie plus probable donné 

 par un syslème de droites non convergenfes. Gomptes Rendus, 1892 I, p. 1415. Journal 

 de I'Ecole Polytechn Gah. 63 (1893), p. 1. 



'") S. FiNSTERWALDER. Bemerkungcn zur Analogie zwischen Aufgaben der Aus- 

 gleichungsrechnung und solchcn der Slatik. Silzungsber. der K. B. Akad. d. Wissensch. 

 zu München, Bd. 33 (1203), p. 683. 



^') R. d'Emilio. Illustrazioni geomelriche e meccaniche del principio dei minimi 

 quadrati. Afli d. R. Instituto Venelo di scienze, lellre ed arti, T. 62(1902—1903), 

 p. 363. 



'^) S. Wellisch. Fehlerausgieicluuig nach der Theorie des Gleichgewichts elasti- 

 scher Systeme. Pantofligek. Fehlerausgleichung nach dem Prinzipe der kleinsten 

 Deformationsarbeit. Oesterr. Wochen schrift f. d. öfï. Baudienst, 1908, p. 428. 



^) F. J. W. Whipple. Prof. Bryan's mean rate of increase. A mechanical illustration. 

 The mathematical Gazette, vol. 3 (1905), p. 173. 



**) M. Westergaard. Statisk Fejludjaevning. Nyt Tidsskrift for Matematik, B, T. 21 

 (1910), pp. 1 and 25. 



^^) G. Albenga. Gompensazione grafica con la figura di errore (Punti determinati 

 per intersezione). Atti d. R. Accad. d. Sc. di Torino, T. 47 (1912), p. 377. 



