ioo PETER GUTHRIE TAIT 



kinds of sciences, then you go back to him, and he tells you a wrinkle. I have 

 done lines of force and = potls. of double tangent galvanometer in a diagram, 

 showing the large uniform field. Is T still in London ? " 



It was in this paper also that Tait proved his neat theorem " that a planet 

 moving about a centre of force in the focus of its elliptic orbit is describing 

 a brachistochrone (for the same law of speed as regards position) about the 

 other focus," or in other words, " while time in an elliptic orbit is measured by 

 the area described about one focus, action is measured by that described about 

 the other." These statements are intimately bound up with the general 

 theorem connecting brachistochrones and free paths already referred to. 



In December 1871 Tait communicated to the Royal Society of Edinburgh 

 a mathematical note on the theory of spherical harmonics (see Proc. R. S. E. 

 Vol. vn, pp. 589-596). The interest of the note lies entirely in the simple 

 manner in which certain fundamental relations are deduced. The article. 

 seems to have taken form to some extent under the influence of Maxwell, 

 who, on a post card of date Sept. 5, 1871, wrote 



" Spherical Harmonics first written in 1867 but worked up from T and T' when 

 that work appeared. Have you a short and good way to find //(^i w ) 2 ^5? If so 

 make it known at ice that I may bag it lawfully as T' 4nion path to harmonic 

 analysis." 



Tait seems to have replied by sending a sketch of a new method, for 

 Maxwell on October 23 wrote (again on a post card) 



" O. T' ! R. U. AT 'OME ? //Spharc'^/S was done in the most general form in 1867. 

 I have now bagged f and 17 from T and T' and done the numerical value of 

 //(Ff'") 2 ^ in 4 lines, thus verifying T+T"s value of tf^MJdS. Your plan seems 

 indept. of T and T' or of me. PUBLISH ! " 



This was followed up ten days later by a fairly long letter bearing upon 

 Tail's notes, the one quaternionic and the other in ordinary analysis. Tait 

 must now have sent his analytical note very much in the form in which it 

 was finally published. Regarding it Maxwell wrote : 



it SCROOPE TERRACE, 

 CAMBRIDGE, 



2 Nov. 1871. 

 O T' 



Your notes have ravished me. An interest in 2<apf being revived this is 

 exactly what is wanted for a quantitative or computative discussion of the symmetrical 

 system considered as depending only on certain symbols * and s. 



