SPHERICAL HARMONICS 189 



the second edition) new ground is broken in the treatment of what had 

 till then been called " Laplace's Coefficients." 



In a letter to Cayley of date February 4, 1886, Tait makes a curious 

 remark which touches on the genesis of this remarkable appendix. He 

 had been poking fun at Sylvester and Thomson for their raptures over 

 new ideas, and concluded : 



Thomson, about 1860, used to lay down the law that the smallest experimental 

 novelty was of more value than the whole of mathematical physics. Then he met 

 with an accident which prevented his experimenting for a whole winter. During 

 this period he became acquainted with Spherical Harmonics, and then his fundamental 

 dictum was wellnigh reversed ! 



Tait of course is having his joke ; but the facts implied are probable 

 enough. Tait's intimacy with Thomson began in 1860 before his appoint- 

 ment to the Edinburgh Chair. Thomson was at the time very busy with 

 electrical work largely the outcome of his labours in furthering the develop- 

 ment of ocean telegraphy ; and Tait must have been impressed with the 

 eagerness of his new friend in all kinds of experimental work. Then came 

 the accident which lamed Thomson for life ; and necessity forced him for 

 a time to devote his energies to mathematical investigations in the famous 

 green-backed note books which were ever afterwards his inseparable com- 

 panions. It is not improbable that the new treatment of spherical harmonics 

 was begun and to some extent developed during these months of 1 860-61. 



In Maxwell's letters to Tait there are several passing references to 

 this section of the " Archiepiscopal Treatise" as Maxwell playfully called 

 it, in humorous reference to the fact that the Archbishops of York and 

 Canterbury were at the time also Thomson and Tait. Thus on Dec. n, 

 1867, he wrote : 



" I am glad people are buying T and T'. May it sink into their bones ! I shall 

 not see it till I go to London. I believe you call Laplace's Coeffts. Spherical 

 Harmonics. Good. Do you know that every Sp. Harm, of degree n has n axes ? 

 I did not till recently. When you know the directions of the axes (or their poles 

 on the sphere) you have got your harmonic all but its strength. For one of the 

 second degree they are the poles of the two circular equipotential lines of the sphere. 

 I have a picture of them." 



Again, on 18 July, 1868, he wrote: 



" How do T and T' divide the Harmonics between them ? I had before getting 

 hold of T and T' done mine for electricity, but I should be delighted to get rid of 



