IV PREFACE. 



as it comes up. The student is supposed to have a fair knowledge of 

 the Calculus, but not of Differential Equations or the Higher Analysis. 

 Many explanations are therefore necessary, some of which are given in 

 the form of notes. 



Two opposing tendencies have at various times made themselves 

 manifest in the treatment of Dynamics, both of which have been very 

 fruitful. Lagrange, in the advertisement to his great work, the "Mecanique 

 Analytique", proudly says, "On ne trouvera point de Figures dans cet 

 Ouvrage. Les methodes que j'y expose ne demandent ni constructions, 

 ni raisonnements geometriques ou meeaniques, mais seulement des operations 

 algebriques, assujetties a une marche reguliere et uniforme. Ceux qui 

 aiment 1' Analyse verront avec plaisir la Mecanique en devenir une nouvelle 

 branche, et me sauront gre d'en avoir etendu ainsi la domaine." Lagrange's 

 boast of having made Mechanics a branch of Analysis has been amply 

 justified by the results obtained by means of his general method for 

 solving mechanical problems, and his pleasure would have been greatly 

 enhanced could he have foreseen the results of extending it to wider 

 fields in the hands of Maxwell, of Helmholtz, and of J. J. Thomson. 

 Nevertheless in attempting to do without figures or mental images we 

 may rob ourselves of a precious aid. Thus Maxwell, speaking of the 

 motion of the top, says that "Poinsot has brought the subject under 

 the power of a more searching analysis than that of the calculus, in 

 which ideas take the place of symbols, and intelligible propositions 

 supersede equations". There is certainly no doubt of the advantge, parti- 

 cularly to the physicist, of having ideas take the place of symbols. 

 The introduction by Hamilton of the notion of vector quantities was a 

 great step in this direction, which has assumed very great value to the 

 physicist, and it was to a particular case of this that Maxwell alluded, 

 namely to the idea of the moment of momentum, or impulsive couple, 

 as it was termed by Poinsot. The importance of this physical or geometrical 

 conception may be seen from the use made of it, under the name of the 

 Impulse, by Klein and Sommerfeld in their very interesting work on the 

 Top. On the other hand this notion of impulse, while in this particular 

 case a vector, is but one case of the general notion of the momentum 

 in Lagrange's generalized coordinates. Will it not then be an additional 

 advantage if, keeping both the analytical and the geometrical modes of 

 expression, we attempt to introduce into Lagrange's analytical method 

 geometrical analogies and terminology? This it is perfectly possible to 

 do, for it turns out, as was shown by Beltrami, and beautifully worked 

 out in detail by Hertz, that the properties of Lagrange's equations have 

 to do with a quadratic form, of exactly the sort that represents the arc 

 of a curve in geometry. Analytically it is of no importance whether the 

 number of variables is more or less than three how natural it is 

 accordingly to employ the terminology of geometry, which must result 

 in giving a more definite image of the quantities involved. For this 

 reason I hope that no physicist will accuse me of having dragged in the 

 subject of hyperspace into a physical treatise.- I have insisted that what 

 is involved is merely a mode of speaking, and has the advantage of 



