TRANSACTIONS OF bEGTION A. Gil 



charges involved in the maintenance and working of an experimental tank at 

 Bushey, and there is good hope of success ; as well as of the readiness of the 

 Executive Committee of the Laboratory to permit the tank to be placed at 

 Bushej'. It may be hoped, therefore, that before long v,e shall have in this 

 country an establishment in •which experimental investigations of wave-phenomena 

 in water may be carried further than has yet been possible; and it must be the 

 conviction of all those who have given attention to the subject that, whether 

 wave-phenomena in water or in the atmosphere be the special subject of study, the 

 foundation of a sound theory must be laid by means of experimental inquiries, to 

 the results of which mathematical processes may be applied in order to establish a 

 comprehensive theory. 



Professor Larmor, Professor Trouton, and JI. Teisserenc de Bort also con- 

 tributed to the discussion. 



Department of Mathematics. 



The following Papers were read : — 



1. Linear Vector Functions. By Sir Robekt Ball, F.B.S. 



The results here given are based upon and are lai'gelv identical with those 

 obtained by the late Professor C. J. Joly, to whom the application of quaternions 

 to the theory of screws is almost entirely due. The author was in intimate 

 correspondence with Professor Joly for many j'ears, and up to the time of his 

 lamented death the subjects now referred to were under constant discussion. 

 The present communication is now offered in Joly's own university as a token of 

 affectionate remembrance. The occasion is the more appropriate as Joly was one 

 of the successors to the chair of Hamilton, whose invention of linear vector 

 functions was one of the most wonderful strokes of quaternion magic. 



Let Xj, <a, ; X.J, w.^ ; X^, <o^ be three pairs of vectors defining three screws. 

 Let <i, t.^, t.^ be the intensities of three wrenches on those screws. These 

 wrenches compound into a single wrench of intensity f on a screw \^ to of the 

 three system, and we have 



^ X = t{K^ + t.X, + t.^.^ ...... (i) 



< <B = t^U)^ + t./O.^ + t.M>.^ . , . . , (ii) 



Multiplying the first by A'AoX^ and taking the scalar 



t SXX^Xj = ijbXj^XjXj, 



by this and the two similar equations we have from (ii) 



to = (cBj^SXXjXj + to^SXX^Xj^ + (B^SXXjX^) /SXjX^X^, 



But the expression on the right-hand side is a linear vector function ^X of the 

 most general type. 



"We thus obtain X, <^X as the two vectors defining in the most general manner 

 all the screws of a three-system. We now show how the properties of such a 

 system are obtained at once as properties of linear vector functions. 



One of Hamilton's most remarkable propositions states that 



(f)Y^'fjL(f)'v = wjY/ii'. 



This is the quaternion statement of the fact that of two reciprocal three systems 

 A and B each screw of A is reciprocal to each screw of B. 



If <^X, X represent a three system, then any screw — wtV/iK, Y(j}fi(f)v will be a 

 screw of the reciprocal system, whatever X, fi, v may be, provided that m is the 

 invariant of the linear vector function defined by the equation 



S<pX<pfx<f)v — mB\[iv = 0. 



bk2 



