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11 



CONTRIBUTIONS TO THE THEOEY OF SCREWS. 



By sir ROBERT BALL, ll.d., f.r.s., 

 Lowndean Professor of Astronomy and Geometry in the University 



of Cambridge. 



Read Jani'aky 10. Onlered for Publication January 12. Piiblislied April 5, 1910. 



CONTENTS. 



Page I Page 



I. On the expression for the Virtiiiil IV. Applications of Quaternions to tlie 



Coefficient of two Vector-Screws, . 16 Tlieory of Screws, . . .44 



! V. Use of Quaternions in the Theory of 



II. f)n the composition of Twists or | Reflected Screws, ... 62 



Wrenches on Vector-Screws, . 27 YI. Quaternion InvesUgation of the Screw 



111. On the Pilch 0[ier»(or 



:r- + 5- -t . . . + — , . .40 



Cpi opt )>, 



Ueciprocal to five given Screws, 55 



VII. Keprcsentation of Screw Systems of 

 the third order \>y Linear Vector 

 Functions, ..... 61 



I. — On the expression for the Virliutl Coefficient of two Vcctor-Screips. 



TiiK properties of the virtual coefficient of two screws are of fundamental 

 importance in that branch of mathematics which is known as the Theory of 

 Screws. The term " virtual coefficient " was firet introduced into the Theory 

 of Screws in a memoir by the present writer.* It must, however, be remem- 

 bered that a certain function of geometrical quantities, which is called the 

 fundamental invariant of two linear complexes, and which is exactly parallel 

 to the virtual coefficient in the Theory of Screws, had been previously 

 employed by Kleinf in a series of important investigations. 



The evanescence of the virtual coefficient of two screws indicates that the 

 screws stand in that remarkable relation which we have expressed by desig- 

 nating them as reciprocal. In the earlier investigations of the Theory of 

 Screws there had been no occasion for the employment of the virtual co- 

 efficient except in connexion with reciprocal screws. But when in a later 



• Phil. Trans. Boy. Soc., 1874, p. 16. 



t Math. .\nn., vol. ii., pp. 198-226 (1869). For further details and references, see the " Treatise 

 on the Theory of Screws" by the present writer. Cambridge, 1900, pp. 17, 617. In future 

 references to this boiik in the piesent paper, it >« ill be terniod "ioiply " Treatise." 



