VORTICES AND KNOTS 105 



Not able to understand the reference to the planet I sent the note to 

 Kelvin himself, who, writing on Oct. 3, 1907, said 



" I return my old pencilled letter to Tait, which has come to me enclosed 

 with yours of yesterday. I have no recollection of the wonderful planet. 



" PFian meant Pecksniffian. Pecksniff was a great hero of Tait's in respect to 

 his almost superhuman selfishness, cunning, and hypocrisy, splendidly depicted by 

 Dickens." 



The only other planetary theorem with which Tait's name is associated 

 is the one already referred to in connection with Action and Brachistochrones ; 

 but this comparison between the effects of cohesion and gravitation when first 

 made was just the kind of thing to appeal to Thomson. 



Tait's excursions into the field of pure mathematics were not frequent ; 

 and his paper on the Linear Differential Equation of the Second Order 

 (Jan. 3, 1876) practically stands alone. It contains some curious results and 

 suggests several lines of further research. The general idea of the paper is 

 to compare the results of various processes employed to reduce the general 

 linear differential equation of the second order to a non-linear equation of 



the first order. The properties of the operators of the form ( x } are 



\8;tr ar/ 



incidentally considered, and the question is asked as to the evaluation at one 

 step of the integral 



At the British Association Meeting of 1876, Tait communicated a note 

 on some elementary properties of closed plane curves, especially with regard 

 to the double points, crossings, or intersections. He pointed out the connection 

 of the subject with the theory of knots, on which he was now about to begin 

 a long and fruitful discussion. He was attracted to a study of knots by the 

 problem of the stability of knotted vortex rings such as one might imagine 

 to constitute different types of vortex atoms. Some of these were figured in 

 Kelvin's great paper, which itself was the outcome of Tait's own experimental 

 illustrations of Helmholtz's theorems of vortex motion. The conception of 

 the vortex atom gave an extraordinary impulse to the study of vortex motion, 

 and the following early letter of Maxwell indicates some of the lines of 

 research ultimately prosecuted by Thomson and Tait. 



T. 14 



