TRANSACTIONS OF SECTION A. 569 



Each optical punctum of this image is the concentrated light of one or of a 

 small group of these beams ; and it can be proved geometrically that the image so 

 formed is a part of the indicator diagram, and possesses the valuable properties of 

 that diagram which were explained in the paper read last year. In virtue of these 

 properties we can, by scrutinising the image seen on looking down the tube, ascer- 

 tain by observation the directions, intensities, and colours of the component undu- 

 lations of flat wavelets into which the light, as it travels from the object under 

 examination to the objective, necessarily resolves itself. This prepares the way 

 for making and interpreting a great body of valuable experiments. 



The full paper will appear in the ' Philosophical Magazine.' 



i. On the Form of Lagrange's Equations for Non-IIolono7nic Systems.^ 

 By Professor Ludwig Eoltzmann. 



If a system is given of n material points, with the rectangular co-ordinates 

 x^x\ . , . x^n and if between the co-ordinates we have i/ equations, the system can 

 be characterised by 3re — 1/ generalised co-ordinates. If some of the v equations 

 have the form ^i<^.i', + . . . ^3n<^.Vyi, - 0, and if r cannot be reduced to the form 

 d(f>{x\x., . . , a.:^„) = 0, then we call the system non-holonomic of the order r. Then 

 also at least r equations between the rectangular and generalised co-ordinates must 

 have the form d.i\='r)^^dp^ + . . . r].^„^^dp.^n-^, and Lagrange's equations are no 

 longer true. Professor Boltzmann develops the members which must be added 

 to Lagrange's equations in this case, and gives a very simple and striking example. 

 The complete paper will be found in the 'Sitzber. d. Akademie in Wieu,' 

 Bd. Ill, p. 1603, December 18, 1Q02. 



Wave-propagation in a Dispersii^e Medium. 

 By Professor A. Schuster, F.E.S. 



6. Diticussion on the Use of Vectorial Methods in Physics.'- 

 Opened by Professor 0. Henbici, F.R. iS. 



Contribution by James Swinbukne, M.Inst.CJ. 



Such a question as this does not concern only the writers of mathematical 

 books ; it is essentially important to physicists. My own unfortunate experience is 

 probably that of very many others. Coming on the notion of quaternions, or at 

 any rate of vectors, in Maxwell, one had recourse to Tait. As this was nearly 

 twenty years ago, Hamilton was not accessible, except in Glan's German transla- 

 tion. After working at an idea that promised great simplicity in the future treat- 

 ment of electrics, with the notion that it would come into use, I found my trouble 

 wasted, for Heaviside's system came along. Then there came all sorts of minor 

 modifications. I have now forgotten all about both systems, and have no intention 

 of troubling my head with any of them in future. Until mathematicians settle 

 among themselves what system is to be adopted, and bury all the others, the 

 ordinary man will not take up any of them. Working with collections of direction 

 cosines and differential equations which have a perspective is very tedious. To 

 follow the meaning of them is great mental labour, and it involves a peculiar sort 

 of imagination, akin to that necessary for playing chess in a far-seeing way, or 

 blindfold — a sort of mental imaging. Whether the Cambridge mathematician 

 pictures these things in his mind, or degenerates into a user of blind mathematics, 



' Printed in extenso in the Sitzher. Ak. derWiss. in Wien, Bd. cxi. 



* Professor Henrici's opening remarks are printed in extenso in the Keports, p, 51. 



