190 PETER GUTHRIE TAIT 



the subject out of the book except in the way of reference to T and T'. My 

 method is to treat them as the neighbourhood of singular points in potential systems, 

 those of positive degree being points of equilibrium, and those of negative degree 

 being infinite points." 



And then followed a brief outline of the method subsequently developed 

 in Electricity and Magnetism, in which the harmonic of the th degree is 

 determined by its n axes and a constant fixing its strength. 



Thomson and Tail's treatment of Spherical Harmonics is essentially 

 physical, their object being to quote their own phrase " the expression 

 of an arbitrary periodic function of two independent variables in the proper 

 form for a large class of physical problems involving arbitrary data over a 

 spherical surface, and the deduction of solutions for every point of space." 

 It is this object which guides them in their analytical work ; and through 

 it all it is abundantly clear that the theory of the potential is ever present 

 to their minds. In no true sense can the appendix be regarded as a 

 sustained piece of mathematical reasoning. The convergency of the series 

 is practically assumed, or, let us say, left to be proved by the reader. 

 But the combined mathematical power and physical intuition are shown at 

 every stage ; and the use of the imaginary linear transformation, a distinct 

 novelty in 1867, leads to an elegant and simple deduction of useful forms. 

 Further on in the book in the sections on statics the authors give other 

 useful developments ; and it is then that they introduce their names, Zonal, 

 Sectorial, and Tesseral Harmonics, according to the character of the nodal 

 circles on the spherical surface. 



Passing on to Chapter n of Division i we find it devoted to the 

 discussion of Dynamical Laws and Principles. One feature of the early 

 sections of this chapter is specially emphasised by the authors : it is a 

 return to Newton. This means in the main two things. In the first place 

 Newton's Laws of Motion are given in Newton's own words and the whole 

 fabric of dynamics is raised on them as the sufficient foundation. In the 

 second place, by adopting the Newtonian definition of force as being measured 

 by the change of motion produced, Thomson and Tait get rid of the wearisome 

 proof of the parallelogram of forces which was one of the marked features of 

 the text- books of the middle of last century. In fact, as Newton showed, 

 the composition and resolution of concurrent forces follow immediately from 

 the second law of motion. Tait frequently remarked that Thomson and 

 he "rediscovered Newton for the world." They also seem to have been 



