On Vector Differentials. 187 



sufficiently thin to allow the a rays to escape, must decrease 

 in weight. Such a decrease has been recently observed by 

 Heydweiler* for radium, but apparently under such con- 

 ditions that the a rays would be largely absorbed in the glass 

 tube containing the active matter. 



In this connexion it is very important to decide whether 

 the loss of weight observed by Heydweiler is due to a decrease 

 of weight of the radium itself or to a decrease of weight of 

 the glass envelope; for it is well known that radium rays 

 produce rapid colourations throughout a glass tube, and it is 

 possible that there may be a chemical change reaching to the 

 surface of the glass which may account for the effects 

 observed. 



McGill University, 

 Montreal, Nov. 10, 1902. 



XVI. On Vector Differentials. By Frank Lauren 

 Hitchcock. — Second Paper f. 



1. rilHE calculus of Quaternions enables us to represent a 

 J- vector, or directed quantity, by a single symbol, 

 and to work with it easily and compactly. We are not 

 obliged to resolve into components, nor do we arbitrarily 

 introduce any lines or planes of reference. 



One of the simplest vector* is that of a point in space, re- 

 presented by the symbol p. If we have a vector function of 

 p continuously distributed throughout a portion of space, we 

 may differentiate it : the result is a linear and vector function 

 of dp, closely analogous, in a mathematical sense, to a homo- 

 geneous strain. Any such strain is fully determined if we 

 know the roots of the strain-cubic, and the three directions 

 which correspond to them. 



In an introductory paper on this subject (Phil. Mag. June 

 1902, p. 576) it was shown that if v be a vector of unit- 

 length normal to any family of surfaces, and if its differential 

 be %dp, then one of the roots of the cubic in ^ is always 

 zero. 



The other two roots give directions tangent to the lines of 

 curvature. For a line of curvature may be defined as one 

 such that normals at contiguous points intersect, that is, such 

 that the three vectors v, v-\-dv y and dp are coplanar ; but 

 because v is a unit-vector dv is at right angles to y, and 

 therefore parallel to dp. Accordingly (%— g)dp = (K g being 

 a root of the strain-cubic. 



* Phys. Zeit. 1902. 



t Communicated bv the Author. 



